You don't have ModBro on your device already?
In general, if you have the link for an APK to download and then install, yes you don't need to use your PC. Open up the Browser or Chrome app on your DroidBOX device, type in the URL (address), and once complete, you'll be able to find the APK file in /sdcard/Download . Use ES File Explorer or File Manager HD to access the Download directory and then click on the APK file. It'll ask if you want to install, and let you know about the permissions the app is asking for.
If you prefer, you can just use a USB thumbdrive, hard drive or memory card to transfer files (such as an APK) that you download on your PC. When you plug the device/card into your DroidBOX afterwards, open up ES File Explorer, and on the left, under the LOCAL menu, you'll see an entry for your card or device. Click it, find your file and
a) Open it as normal if it is an APK you won't need again in the future
OR
b) To transfer the file, long-click on it. One of the new options at the bottom will be COPY, select that. Navigate to somewhere on your device, perhaps the download folder, then click the PASTE button at the bottom.