First, make sure that you are following all the necessary principle for performing quantitative Western blots.
Second, you say:
I want to perform a statistical test to compare the means of the two groups to see if there is a significant difference. I am planning on using the Mann-Whitney U test since I do not want to assume that my data is normally distributed.
The Mann-Whitney U test does not compare the means of two groups. See the Wikipedia entry:
The Mann–Whitney U test tests a null hypothesis of that the probability that a randomly drawn observation from one group is larger than a randomly drawn observation from the other is equal to 0.5 against an alternative that this probability is not 0.5...
That said, the Mann-Whitney can be a very good test of whether 2 sets of unpaired observations differ in that way.
Third, you can perform a statistical test with a limited number of samples. The question is whether a limited number of samples can provide the statistical power to find that a true difference between two groups provides statistical significance, given the variability among the measurements. To evaluate the number of cases to provide adequate power, you need to have an estimate of the variability of the measurements and the magnitude of the true difference you would like to be able to detect.
There are ways to do power calculations for the Mann-Whitney test, explained for example on this page, based on those principles. I suspect, however, that for power analysis you will be better off using the information you have about the variance of Western blot measurements within treatment groups as a guide, calculating the power based on an assumption of within-group normality. See this answer for some ways to do that.
Finally, my experience with Western blots is that reviewers will be most convinced if you show visually distinct differences between samples of the 2 groups that were run in parallel on the very same gel and then blotted and stained together. Even if you demonstrate a "statistically significant" difference between the groups based on separate Westerns, a reviewer will tend to discount that finding if she can't actually see the difference on a single blot.
There's a good reason for that insistence on visual demonstrations of differences between samples from the 2 groups: the same sample run on different gels and blotted for the same protein on different days can show very high variability in staining intensity, even when normalized against a loading control (e.g., actin, tubulin).
With such variability among Westerns, paired analysis is best both for visual and for statistical discrimination. In particular, you could then do paired comparisons based on each pair of samples from the 2 groups, run on the same gel and blotted/stained together.
Unfortunately, as you have described your work to date, you didn't do this pairing within Westerns. I strongly suggest that you re-do your analysis by pairing within Westerns, whether you are doing statistical analysis or not. It's the best biochemical practice.