I randomized samples into two experiments and I ended up with about 190 samples in one group and 80 in the other. The distributions of the two samples are not normal and I want to use Wilcoxon test to see if the treatments are identical. Should I pick equal samples from both groups or is this requirement waived off for non-parametric tests?
It depends on which Wilcoxon test you are refering to. The Wilcoxon signed rank test is for paired data. From what is sounds like you do not have paired data so this would not be the appropriate test. The Wilcoxon rank-sum test is for two independent samples and does not require equal sample size for each group. However, there are assumptions that come along with this test (as any non parametric test does-- just less than the parametric version) so I would suggest looking those up beforehand.