Yes it is perfectly possible to do this either in Stata or in R. Since I use R I offer a few hints to get you going.
A list of software for meta-analysis in R is available in the CRAN Task View (Disclaimer, I maintain it). There are several packages there which will do what you are proposing. I personally use
metafor but there are other options.
You will almost certainly need to choose a transformation for your proportions before doing the meta-analysis and then back-transforming for interpretation. This is so your estimates are more approximately normally distributed. I would suggest the logit is worth considering. If you have any zeroes you will need to deal with them by adding a constant.
Adding moderator variables presents no new problems but with 20 primary studies you cannot add too many at once.
After fitting there is a range of graphical techniques you can use to display your model. Forest plots are commonly used for this but there are others available.
And finally it is worth, if you do use
metafor to check the author's web-site which has many examples and hints. Particularly relevant may be these two here and here