First I'll state that I have Admin rights on the laptop I use, though I've helped a co-worker get the same setup going on his machine and he didn't have Admin rights to his machine. And by that I mean as our company has his non-admin account defined as, not as your company may have setup.

We just downloaded the appropriate Basic Light Package and SQL*Plus Package from Oracle. For 32-bit, it would be https://www.oracle.com/database/technologies/instant-client/microsoft-windows-32-downloads.html We unzip the two files into a directory we created (I use C:\Oracle but anything can be setup where you have privs to create a folder). Edit Environment Variables to add in an ORACLE_HOME to point to C:\Oracle\instantclient_x_y, edit Path to add in this same directory, create a TNS_ADMIN entry to point to C:\Oracle (this is also where I have my tnsnames.ora) file. Start up PL/SQL Developer and it works. I make those changes in the System environment variables, but I suppose they should also work if made for just environment variables for my account.

For your issue, I don't think removing PL/SQL Developer is necessary, but maybe wiping all the Instant Client software and verifying all traces of them are gone.