Can "Center" provide VFR flight following?
Any US radar facility can (and will, workload and coverage permitting) provide VFR Flight Following, whether Approach/Departure or Center.
Look up the airport you are departing or, if already airborne, the nearest airport in your general direction of flight in your A/FD, EFB or (if you’re lucky) glass panel. The App/Dep lines will give you the frequency and name of the facility to call in that area.
The coverage problem is mostly an issue of altitude rather than range. Center radar and radio sites are further apart, so due to terrain and the curvature of the Earth, they may not be able to see/hear you if you’re too low. Typically 3,000 AGL is high enough, but if not, just keep climbing and try again in a few minutes.