Yes there are many aquatic plant species which do not stand up in water.
You need to look at the lifeform of the plants, such as introduced by Raunkiaer ( https://en.wikipedia.org/wiki/Raunki%C3%A6r_plant_life-form ) and later work by Ellenberg & Mueller-Dombois (1967).
In fact, you're mixing up two different lifeforms :
- true aquatic plants are unable to "stand up" : they are waterborne so they "bend over" depending of the water currents. You'll find rooted aquatic plants as well as rootless ones which float free above or within the water column.
European examples of rooted true aquatic (freshwater) plants are in the families NYMPHAEACEAE, CERATOPHYLLACEAE, HALORAGACEAE, RUPPIACEAE, ZOSTERACEAE, POTAMOGETONACEAE, ZANNICHELLIACEAE, CALLITRICHACEAE, NAJADACEAE and RANUNCULACEAE (Ranunculus subgen. Batrachium)
- helophytes is the name of plants which stand in water (usually have their roots and the basis of their stalk within water) but are erected and have aerial part standing up outside the water
European examples of helophytes are : Sparganium, Typha, ALISMATACEAE, many JUNCACEAE and CYPERACEAE, some POAECEAE (Phragmites, Glyceria, and a long etc.), and many other plants in many families.
For european France, Philippe Julve gives a comprehensive list of all plant species with mention of their life form, so you can look up species which correspond lifeform of your interest. Look here for the "CATMINAT" folder and then baseflor.xlsx and look in the first tab the column R "Type_biologique" (which means life form).
You will want to look the meanings of the abbreviations in the "Légende" tab of the sheet, column G. Julve consider true water plants as a sub-category of other main life form, so you'll want to search for "something-aqua".
Since you're interested in aquatic vegetation, you may want to look at some phytosociological catalogue like E-VEG (on the left side you can browse the tree to section "1 Marine waters" and "3 Freshwater aquatic vegetation".
Clic on one sub-category (latin names of vegetation units) will open the corresponding record in the main frame where you'll find indication to some litterature. But it may be of more interest for you to browse some academic catalogue for the latin name of the higher vegetation units + name of your area of interest. For instance you may want to look for Stuckenietea pectinati + Europe.
Be aware that E-VEG choose to follow botanical nomenclature (so when the main species after which the vegetation unit is named change its name, then the vegetation unit's name change as well, so when Potamogeton pectinatus became Stuckenia pectinata then the Potamogetonetea pectinati became the Stuckenietea pectinati). But many other phytosociological catalogues choosed to conserve the vegetation unit's names stable regardless of plant name's changes. So you may prefer to look for Potamogetonetea pectinati + South-East Asia (if that's your area of interest).
If E-VEG don't suit your needs, you may want to take a look at other catalogues referenced here.