Short answer
All marine life needs energy to survive and reproduce. "Heterotrophic" organisms get their energy from eating other organisms and digesting the molecules of their tissues for driving cellular respiration/function. Ultimately, though, entire ecosystems and food webs are dependent on the non-biotic energy (e.g., sunlight) captured by "autotrophs" (aka "primary producers") and stored in the molecules of their bodies.
Plants make up only a portion of the autotrophs present in the oceans. Photosynthesis by phytoplankton accounts for the majority of Net Primary Production (total energy stored by autotrophs minus energy used by autotrophs to live) in our oceans. Chemosynthesis of bacteria and archaea in the ocean depths account for a small degree of NPP, too.
Heterotrophic organisms eat the autotrophs, and higher trophic heterotrophs eat lower trophic-level organisms. Dead matter (still containing energy-rich molecules such as sugars and fats) drift to lower ocean depths to provide organisms farther from the photosynthetic surface regions with most of their energy. At all trophic levels, energy is used to drive cellular work or lost as heat; as such, less and less energy is available at higher trophic levels and therefore necessitates eating more and more food.

Illustration by Tim Gunther. Source: National Geographic
SUMMARY: all marine organisms are dependent on primary producers, most primary producers are photosynthetic, plants are responsible for a significant (though minority) amount of productivity in oceans, and phytoplankton account for the majority of energy capture in our oceans.
Long answer
The Basics
All biological systems on earth require energy and matter for survival. Primary producers are the organisms that acquire energy from non-biological sources to make them available to the rest of the biotic component of the ecosystem (i.e., their associated food web; sometimes visualized as an energy pyramid).
Most commonly, primary producers (or autotrophs) acquire energy from the sun, and therefore are referred to as photoautotrophs. Other organisms, however, acquire energy from non-solar sources (e.g., thermal vents and cold seeps at the ocean bottom), and are termed chemoautotrophs.
If all other organisms in an ecosystem depend on these primary producers as an energy source, it's important to try to quantify how much energy the autotrophs are converting from non-biotic sources and storing in their cells. We can quantify this in terms of productivity:
Gross primary productivity (GPP) is a measure of the conversion rate of the above non-biotic energy sources (typically solar) into stored chemical energy (typically in sugar molecules) by the autotrophs.
Net primary productivity (NPP) is essentially GPP minus energy used by the autotroph to survive (i.e., respiration).

Read here for more information about energy flow basics.
Primary Producers in Oceans
Although photosynthetic plants are the best known primary producers, they are not the only autotrophs on earth. In fact, the large portion of ocean productivity is a result of phytoplankton, a diverse collection of photosynthetic protists and cyanobacteria.

Image source: NASA, originally from Sally Bensusen, NASA EOS Project Science Office
The photosynthetic activity of plants and phytoplankton provide the majority of NPP for the entire ocean, but they are both limited in where they are located.
Sunlight is limited by water depth, reaching depths of less than 300 m.

Image source: NASA
Since sunlight-availability dictates NPP to such a large extent, we can divide the ocean into vertical zones based on light availability. Only the top 200 m, referred to as the euphotic zone, has enough sunlight available for photosynthesis.

Image source: NOAA
Both of these photosynthetic organisms are also limited by nutrient availability.
For example, phytoplankton are most abundant in upwelling zones where sinking nutrients are brought back up to the ocean's surface. These occur both in the open ocean and along coastlines.
Plants, which absorb nutrients through root systems, are further limited by the availability of substrate within the euphotic zone. In other words, they need somewhere to "anchor" down. As such, plants are only found along continental shelfs or other shallow benthic or sedimentary regions within the euphotic zone. In other words, we would say plants are limited to coastal or littoral zones.
Although the majority of production in oceans is a result of euphotic photosynthesizers, very small amounts of production are the result of chemosynthesis at thermal vents and cold seeps in deeper ocean depths. the chemoautotrophs here are all bacteria or archaea.
NPP of our Oceans
So far we discussed the importance of net primary production (NPP), the autotrophs responsible for the ocean's NPP, and the limitations of where such autotrophs can live. The question still remains: to what degree do the oceans rely on these variable sources of NPP?
According to Fox et al. (2020), "Assessments of marine primary production...are made difficult by the extreme plasticity in phytoplankton physiology
". However, certain broad trends are apparent.
Well, given that both plants and phytoplankton can exist in coastal/littoral zones, these zones (especially those associated with estuaries bringing in additional resources from freshwater systems), tend to have the greatest NPP per unit area of anywhere in the world.
However, since the pelagic, or open-water, portions of the euphotic zone make up such a large percentage of the earth's surface (~70%), this so-called epipelagic zone accounts for more NPP per year than any other biome or life zone globally just due to the immense area/volume for phytoplankton to live and photosynthesize.
- Although phytoplankton make up <1% of the earth’s photosynthetic biomass, Field et al. (1998) estimated such phytoplankton are responsible for almost half the world's annual NPP.
George Kling (University of Michigan) provides 2 simple graphics demonstrating these two points on a course website of his here. I've copied lower quality images here for reference:

Modified from figures 4-5 of George Kling (University of Michigan) course page
Since only autotrophs can convert non-biologic forms of energy into chemical stored energy that heterotrophs such as fish and other water creatures can consume, all non-photosynthesizing organisms throughout the ocean's depths rely on the consumption of these autotrophs (or lower trophic-level heterotrophs) for energy. In fact, according to sigman and Hain (2012), usage of the energy captured but autotrophs and utilized by higher-trophic heterotrophs is so robust that all but <1% of the energy initially captured ever reaches the ocean floor as fallen debris/wastes/decomposing "snow".
As mentioned previously, a minor degree of chemosynthesis along the ocean floor exists, but such chemosynthesis adds minimal NPP to those zones. I could not quickly find a source that quantified the degree to which NPP of these deep zones comes from chemosynthesis (though see McNichol et al. (2018) for a fairly recent publication discussing and examining this topic). However, such chemosynthesis can only occur at thermal vents or cold seeps and therefore are quite limited in their spatial scope. Suffice it to say, that a very limited amount of NPP come from such chemosynthesis and the degree to which chemosynthesis adds to the NPP is spatially variable based on the presence/prevalence of inorganic energy sources such as thermal vents and cold seeps.
Further reading:
Chapter 17 "the Flux of Energy through ecosystems" in Begon, Townsend, and Harper. 2006. Ecology: From individuals to ecosystems. Blackwell Publishing, Malden, MA, USA.
Sigman, D. M. & Hain, M. P. (2012) The Biological Productivity of the Ocean. Nature Education Knowledge 3(10):21 [see here]
Citations:
Field, C. B., Behrenfeld, M. J., Randerson, J. T., and Falkowski, P. G. (1998). Primary production of the biosphere: integrating terrestrial and oceanic components. Science 281, 237–240. doi: 10.1126/science.281.5374.237
Fox, J., Behrenfeld, M.J., Haëntjens, N., Chase, A., Kramer, S.J., Boss, E., Karp-Boss, L., Fisher, N.L., Penta, W.B., Westberry, T.K. and Halsey, K.H., 2020. Phytoplankton Growth and Productivity in the Western North Atlantic: Observations of Regional Variability From the NAAMES Field Campaigns. Frontiers in Marine Science.
McNichol, J., Stryhanyuk, H., Sylva, S.P., Thomas, F., Musat, N., Seewald, J.S. and Sievert, S.M., 2018. Primary productivity below the seafloor at deep-sea hot springs. Proceedings of the National Academy of Sciences, 115(26), pp.6756-6761.
Sigman, D. M. & Hain, M. P. (2012) The Biological Productivity of the Ocean. Nature Education Knowledge 3(10):21 [see here]