I looked up all I could on Land Rovers and got a bunch of information. But I haven't been able to find out exactly when the Rover 7 ambulance came out. Anybody know precisely when? Thanks.

The rover came out in 1961. have a look at:

http://www.eamodels.co.uk/tamiya-british-ambulance-rover-7-kit-135-scale-3264-p.asp

It is the series 2A the kit is based on.

Thanks. One thing that gets me is that once Tamiya had the tooling, they didn't just go ahead and make more versions of the Land Rover while they were at it.

..they did, The SAS Pink Panter was one version, I think this one Was the original Landy mold, this then led later to the development of the ambulance version.
Italeri shared the same mold producing they're own SAS version (with different tyres), around the same time they produced a basic civi soft top version with optional front bumpers for a military variant (no markings).
Later revell did the same & produced another basic soft top using (it appears) the same mold but marketing it as military variant, mostly British including gulf war markings, how accurate it was is another debate