If you buy/rent a motorbike in Vietnam do they have ones with the
abilities to strap on a large backpacking backpack?
A lot of South Asian countries have a collection of powerful pre/post war bikes which carried on production locally after the wars ended. Those were very accomadating bikes. For ex, does this (googled) bike's setup offer you enough storage space ? http://img.photobucket.com/albums/v204/icarus2607/Sikkim%202007/Picture032.jpg
If that wasn't enough, you can also opt for a motorcyle with a side car : http://databikes.com/imgs/a/d/b/r/x/royal_enfield__bullet_350_team_1986_2_lgw.jpg
Such bike mods are pretty common in many South East Asian countries (Pakistan, India, Nepal, Bangladesh, Thailand, Burma etc). Unlike in US, many people over there travel luggage heavy on motorcycles and finding such a bike for rent shouldn't be a problem in a tourist place.
I've seen just regular motorbikes without any attachments for carrying
luggage, but I'm assuming if two people can ride a bike then there's
no reason a single person couldn't travel with a bag right?
Oh yes, there should be no problem at all, from my experience in Asian countries so far. For specific local laws you will have to look up their motorcycle law manual. But as a general observation, it wouldn't matter.