If I apply for my Thai Tourist Visa in Vientiane, say on a Tuesday, will I be able to retrieve it on Wednesday ?

Yes, unless Wednesday is a holiday in Laos or Thailand and application was made in the morning.

Good Idea to get there early, the queue starts about 9.00am..and the application is only accepted until 12.00 noon..you can pick up your passport the following day at 1.00pm onwards...even if it's a long queue is say 100 mtrs....of humans, it will only take about 25 minutes to get your passport back, then go to the House accross the grass in the compound and squeeze into the small waiting room for your name to be called to pay the fee,then your done.

If I apply for my Thai Tourist Visa in Vientiane, say on a Tuesday, will I be able to retrieve it on Wednesday ?

As others have said, yes, should be no problem if you get it there Tuesday AM. And nevermind the signs posted around the queue that say it takes 2 processing days.