I have a response variable that lies between 0 and 1. It is a proportion. My dependent variables are categorical. Normally, I would run a two-way ANOVA but, in this case, would it be okay to run a beta regression? Or is there a two-ANOVA equivalent that I can run?

Beta regression is fine for this. If the mean value and the majority of your percent DV is between 0.2-0.8, still running linear regression can be OK in some scenarios. The issues arise when values and precession estimates are close to the bounds (0 or 1).

Likely, beta regression is a good fit. You can also plot the dependent variable using a histogram to visualize how close it is to '0'. If I recall, the interpretation of coefficients from beta reg is different from linear reg, so keep that in mind if your use it.

What is your sample size and what is the mechanism for so much missingness?

The sample size is 28 observations (very small). One suggestion I have had is to run two-stage or two-part models that can be used for data with excess zeros. That has led me to look at running a Zero-inflated Poisson regression. It is used to model count data that has an excess of zero counts.

Correct, for "count of a certain type of cell divided by the total number of cells" can be modelled with a count model. Do you have zero-inflated data? There is also the negative binomial model.