I'd personally want to use an establishment or person that works exclusively on RV house repair, restoration, or rebuilding. I'm sure a person installing carpet in a stick house would tell you that they'd do it but do they know the intricacies of RVs? ...slides and other structures or characteristics exclusive to RVs?
Here's a mobile repair guy in Burlington that I've heard from others does good work (although I've never used him myself) and I'm sure he could install carpet for you:
Vermont Road and Home -- Mobile RV Service
I'm sure there are many more places that do RV house repairs around Burlington that can also do carpeting.