How can I install apps from unknown sources using my Google account on the web?
If your device is rooted, you can create Install Job in SureMDM.  

Step 1: Create an free account here.
Step 2: Install Mobile agent here.
Step 3: Login to Web admin console here.
Step 4: Click on Jobs on Utility panel.
Step 5: New Job > Android > Install program > Job name > Local File path.
Step 6: Apply the job.
Step 7: You can verify the status of Install Job in the System Log.
It works