Neo-Anarchists Guide to Real Life has a chapter about Doc-Wagon, its 1st or 2nd ed shadowrun, so may not be that easy to get hold of, it does however cover some things worth noting for a campaign like you describe. This may actually also be interesting for GM's running normal campaigns.
1) Doc Wagon contracts will rarely cover shadowrunners - they are not going to bail you out when you get stuck in a firefight on Ares land. They do have deals with some corps, but mainly, they operate on national soil, not corp land.
2) Getting a DW contract, requires a tissue sample to be stored with DW. Something I can easily imagine most shadowrunners with a little sense would instantly categorize as a deal breaker, especially awakened characters.
3) Even in the worst of situations (HTR, High Threat Response) DW employees only fight DEFENSIVELY, their job is to get the customer out of the situation, their job is not to kill the opposition unless its nessecary, and if its a standard response and not a HTR, they may even stand back and watch, only going in to their customer after the situation is secure for their personel.
4) Its business, if a bomb goes off and 5 people are near death, only the one with the doc wagon contract will be saved.
Personally, Id say a Doc Wagon campaign leads to alot of problems. First there is the fact that every mission is roughly the same - Go to a location, extract contract holder, dump him/her at a Doc Wagon clinic. Sure it can be varied a little by having gangs place false calls trying to steal gear, or the moral dilemma of ignoring fatally ill people who covered by a doc wagon contract, or save them and risk unemployment. But generally speaking, I think the pool of plot ideas will dry out fast, many of the aspects which makes shadowrun unique and great (in my opinion) is also void in this type of campaign, no legwork, not much need for anything but combat, vehicle and medicine skills, little actual social interaction. It would basically be all about combat - which I suppose is fine for some but not my taste

The section about DW in the Neo-Anarchists guide to real life is about 9 pages long, and most of it is shadowtalk - as a gamemaster you would (imho) need to make up alot of stuff - standard operating procedures etc, but then you could skip on most of the work in a standard campaign (figuring out subtle plot lines, legwork, contacts etc).
I could see it work as a start of a compaign, 2-3 missions - with some added trouble brewing (cut backs, moral dilemma's etc) all ultimately leading to the characters quitting and joining the shadows as regular shadowrunners, but I wouldn't recomment trying it as a long term campaign idea.