There are now (as on December 21) five different nationally approved covid-19 vaccinations being administered worldwide. These are being manufactured by Pfizer and Moderna (US), Sinovac and SInopharm (China), and Gamaleya's Sputnik vaccine in Russia.
So, in theory at least, a person could opt to take more than one vaccine.
In practice, though, this is not going to be possible any time soon, as there is a centralised registration process, and a subsequent queue in all countries. It is very unlikely that an individual would be able to get approval for taking more than one vaccine.
There are over 200 vaccines in development worldwide using eight different technology platforms, and it is possible that there would eventually be unrestricted access to them.
Not much is known about the benefits or dangers of taking more than one vaccine, as such studies have yet to be done. However, scientific opinion suggests that multiple vaccines may actually increase the body's ability to fight off the virus. This has been likened to booster shots given for several other vaccines. Also, the variations in technologies being used could be an added advantage.