How can I reinstall Play Store?
Of course you can install all your applications via their .apk package, downloadable from the internet (you'll also find several alternative markets like aptoide and F-Droid).
As of play store: the website only tells the android app to install a certain app, so if you don't have it installed, you won't go any further. 


install play store via it's .apk  there
if, once it is installed, it opens and then closes immediately, it is possible that the company that fixed it flashed a firmware without it. In this case, you'll need to flash the appropriate GApps package


PS: I highly recommend to flash gapps as most apps have google-modules dependencies (for instance google play services)