I know, this topic is not Visa related. But i guess in this section i have the best odds to gain a hint.

 

All the shops i asked wanted to see my passport. May be someone know a reliable shop, which rent out a scooter for a copy of passport / deposit. A standard 125ccm automatic would be fine.

 

maybe walk /taxi public transport for a couple of days

Highly recommend renting a scooter or motorbke in Vientiane - there are some very interesting areas, that most visa runners never get to see. I don't remember the shop, but I do remember having to visit 2 before I found a marginally useful bike.  In fact I have rented a scooter in Vientiane, Luang Prabang and Savannakhet.  Surprised they insist on a passport which I refuse to leave.  Sometimes I get away with just my CAN Drivers Permit, and no deposit, Just let them know your Passport is at the embassy and if they still seem reluctant, that your friend rented with them a year ago without his passport and it wasn't a problem.  Just make sure you get a decent bike; many scooters for rent in Laos are ready for the scrapyard.

IMO a bicycle is a better choice for getting around Vientiane town. Some hotels may provide them for free. A scooter is good for further ventures.