There are definitely companies that do this, my company has been providing virtual sysadmin services since 1997. So I'm obviously biased there and won't make any recommendations. There are other options as well:
- Managed hosting: Many hosting companies also provide additional services of various levels. Everything from the basics like performing regular updates, configuring and managing monitoring, being first responders to alerts, managing certain configurations or all custom configurations of system software... My company also does this, so it'd be a conflict for me to make any recommendations there.
- Often it's possible to find a sharp high school or college student that has the passion for sys admin and can work part time. Obviously they will have more limited experience, but depending on your needs this may work well during development and initial beta launch at the least.
- Check your local Linux Users Groups to find someone in your area that may be doing this sort of consulting work, or may be between jobs. When we started out, we found our first two clients through the local LUG and jobs mailing list.
- Search on google for "linux consulting" brings up quite a few paid adverts. Unfortunately, my person who does our google advertising is out now so I can't ask what keywords she would recommend searching for.
- Try sending out a request for this sort of consulting to your LinkedIn networks. I've found it very handy in the past, for example when I needed recommendations for a terrestrial business wireless provider in an unfamiliar city. I got excellent responses from people a few degrees away from me.
In short, there are definitely companies out there that do this sort of work. One recommendation I would make is that unless you really need someone on-site for work (say, you need a lot of assistance with physical jobs in your offices), that you consider broadening your search to outside of your geographic area. You may find a better match that way, and if it is about web-based services it often is done remotely anyway. For example, the vast majority of our clients are out of state.