Is it possible to do a Binomial hypothesis test of 2 samples with widely different sample sizes? I have two samples I'd like to do a hypothesis test on. The sample proportions are:
$$
\hat{p}_1 = \frac{120}{4000}
$$
$$
\hat{p}_2 = \frac{9000}{300000}
$$
If I did a test of two proportions where the null is $p_1=p_2$, would there be issues because $n_1=4000$ and $n_2 = 300000$?

Similarity in sample size between groups is not an assumption for any hypothesis test for proportions that I'm familiar with. You should be fine to run the test.
Note: if you are planning a study, the study will be better powered (there will be a higher probability of finding a result if there is one) if the sample sizes in the two groups are similar. You don't want to plan a study and recruit 10 people in one group and 1000 in another group.
