At the risk of further confusing the issue (there are already 2 answers focusing on different aspects!!), I am going to focus on a somewhat different aspect of this question that I think we need to nail down before it can be addressed.
Going mostly off your question title, it would help to know what you mean by genotype vs. allele frequencies. These are quite different, and it's not clear how you are trying to employ them. Speaking strictly, we use alleles to talk about genetic variation within a locus, and genotype variation to also mean variation across loci as well (how alleles across loci are sorted into individuals; this is sometimes also called "gamete frequency"). Of course, people generally talk about HWE in the context of only a single locus, which means that it is easy to get tripped up here.
It is possible for us to have non-equilibrium genotypes while also having equilibrium allele frequencies, in direct response to your question title. A trivial example of this is linkage: when two loci are very close to each other on a chromosome, their alleles will be more closely correlated than genes on different chromosomes. This yields the phenomenon with the unfortunate name linkage disequilibrium (LD), which implies nothing about evolution or HWE but rather about a non-equilibrium mixing of alleles into genotypes.
For context, it is quite common to apply tests of HWE locus-by-locus in genomics, as a sort of quality control or preprocessing step. It is also common to measure LD for relevant pairs of loci, but there is not much biological relationship between these measures (even though LD can be thought of as an extension of Hardy-Weinberg to a multilocus case, for historical perspective see here.). In any realistic case, global linkage equilibrium across the genome is very unlikely, whereas HWE for most loci is very likely.
As a thought experiment, imagine that a (neutral) inversion arises, and then starts segregating. All of the alleles remain at HWE, but suddenly you have new LD (genotypic disequilibrium) that wasn't there before.
With all that out of the way, I see no reason why evolution could not happen at the level of overall genotypes even in the case that HWE is maintained statistically. This can be achieved by the evolution of e.g. sign epistasis, I am sure there are other examples as well that would yield apparent HWE while trait evolution is technically still going on.
I think that it might be fair to say that monogenic evolution isn't happening with apparent HWE, I don't think that you can say the same thing for polygenic evolution.
I'll finish by just saying that I refer specifically to the statistical phenomenon of HWE, and not to the assumptions themselves (which would be tautological; "can evolution happen if you assume there's no evolution?" as commenters point out). After all, You can violate most of the assumptions of HWE and still end up with allele frequencies that look like HWE:
Although statistical deviation from Hardy-Weinberg expectations generally indicates violation of the assumptions of the theorem, the converse is not necessarily true. Some forms of natural selection (e.g., balancing selection, which maintains multiple alleles in a population) can generate genotypic frequency distributions that conform to Hardy-Weinberg expectations. It may also be true that migration or mutation is occurring, but at such low rates as to be undetectable using available statistical methods. And, of course, all real populations are finite and thus susceptible to at least some evolution via genetic drift.