Hi Ed14, welcome to Android Forums.
While it is possible, you won't be able to do it from the play store. Trying to download apps from Google's Play store is a violation of the terms of service, and we don't condone that sort of thing. There are other sites that can allow you to download the .apk file that you'll need to install an app. Some are legitimate like getjar or perhaps the Amazon app store (I don't remember if they permit downloading app files). There are others that are not so legitimate. The offer all kinds of apps for free and turn out to be loaded with malware, or violate the dev's rights. I can't NOT recommend those strongly enough. It's just not worth the risk.
If there are specific apps that you want (from the play store, or other locations) what I'd recommend is contacting the developer directly and explain your situation. They may be willing to shoot you a copy of the app or they may have it posted on their own website. Most developers are very willing to accommodate users. (God bless 'em).
Now, once you have the .apk file ...
You will need a file manager to install the file like Astro or ES File Explorer. For this one you will probably have to bite the bullet and use your mobile data because even if you can download an .apk for a file manager, you need one to install it. Catch 22. You will also need to go into menu>settings>security and enable unknown sources.
Copy the file from your PC to your phone and find it using the file manager. Depending on which file manager you chose, it should be as simple as selecting the file to install it.