You can but you have to find a means of 'side-loading' the APKs. One way or another you have to get hold of the actual APK file and get it on your Blackphone.
One way I do it, is I use Raccoon on a PC to download the APKs to a folder. Then I have that folder auto-sync to my phone with BT Sync. I then open the APK file from within the BT Sync client on my phone and voila!
Also note that other app stores besides Play do work natively on the Blackphone, such as Amazon or F-Droid. F-Droid only has open-source apps, but Amazon offers a lot of the same apps that the Play store does.
It doesn't matter where you put it, when you put it in the folder of your choice, just click on it and the setup begin. Once the setup is done, you don't need the .apk file anymore so you can delete it. Yes, once the app is install, it will appear in your apps section. I think that when you install an app from the play store it works the same way except that you do it manually this time. Also, I had some troubles with some app, sometimes I download the .apk file and when I click on it on my phone it doesn't work. If I try to download the .apk file another day it works. Just so you know.
EDIT : It is written in another Thread but in case you didn't find it, the Open source app store that /u/privacypatriot is talking about is called F-Droid (https://f-droid.org/). They got an app-store that you can install on your phone so you don't have to deal with .apk file with this one.