Do HVAC contractors do "just" chimney lining/B-vent work

I have an ongoing chimney issue. My existing 10-foot above-the-roof-line masonry chimney has to come down because it is in bad shape. I would like to take it down to near the roof line (to maybe 1 or 2 feet above the roof line), and then have the remaining masonry chimney relined and B-vent added from the roof line up.
Someone suggested that I should try calling HVAC contractors and ask them to just do the chimney relining etc. without changing or doing any work on my heater. What they would need to do is add a flexible metal liner inside my existing masonry chimney, connect it to a metal double-wall "Type B" ("B-vent") chimney vent coming out of the top of the chimney, add a B-vent cap on top of that, and point/repair the 2-foot masonry chimney above the roof line.
Is this something that HVAC companies could or would do -- including getting the permit for the work, etc?
I know that I can just call some HVAC companies and ask, but I thought that I would check here first. I never heard of HVAC companies doing this kind of chimney work unless it was part of a job that involved repairing or replacing a heater.
P.S. The existing heater and hot water heater are gas-fired units, so Type B ("B-vent") should be fine.
P.P.S. I also have an ongoing issue regarding how high off the sloping roof line the B-vent needs be according to code. Some people are stating that the rule is that the top of the B-vent needs to be at least 2 feet above the point where the B-vent is horizontally 10 feet away from the sloping roof. But, for natural gas appliances (which I have), when using Type B vent less than 12 inches (mine would be 6 inch B-vent), then the 2006 International Residential Code for one and two-family dwellings (mine is a one-family), Section G2427.6.4 on Page 382 would apply.
Here is the link to the 2006 International Residential Code for one and two family dwellings: .
Check out Page 382.
In my case, with a 12/12 slope, In think my B-vent would only have to go up about 4 or 5 feet from the roof line.
Is that correct?
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