Keep it simple, if it is only you making changes and your primary business is selling on the site and not the software then just use something like rsync or ssh/scp etc and script it. At most use something like ansible, chef or puppet.

If your primary thing is the software and you are deploying many times a week, then it can make more sense to have more sophisticated controls, but even then as one person a few scripts is usually plenty good enough.

If your primary business is the products you sell and not the software, I would be curious why you chose to build a shopping cart/e-commerce solution. There are so many good solutions out there it is usually just a distraction to build your own unless you have hit a large scale or are in just such a custom niche that nothing works for your product/service unless you write it.