Unfortunately this is a very "yes and no" type question. The system can be converted but there are a lot of variables to consider, and the bot***** *****ne will always remain that the system will not perform at original standards.
Any R12 system can be converted to R134A as the general principle of how the two systems work is identical. Changing the refrigerant in it is the only variable and both refrigerants work more or less the same. For durability however, the simplicity ends there.
First off the R134A is not compatible with the oil used in R12 systems. This means in order for the compressor to last as long as possible, you need to remove it and fully drain the oil from it and replace it with the appropriate PAG type oil for R134A. If that was not bad enough, the condenser, evaporator and drier all have oil in them too (As well as small amounts in the lines). If you want to do it right, these components need to be removed and drained or flushed out with an AC machine to remove all old oil. This is pretty much where just about everyone stops with the conversion process as it begins getting very expensive since few people have proper AC flush machines (not just a recovery machine, not that they would contaminate it on an R12 system anyway) and the labor to pull all those parts for manual draining/cleaning is very high.
So to that end, most people will just evacuate the system or at most pull the compressor and drain it then recharge with R134A and PAG oil. This is better than just putting the refrigerant in, but will still take a toll on the compressor in the long term as the other oil in the system circulates around.
The other big issue with difference in the gases is the pressure at which they operate. R134A operates at a higher pressure than R12, with typical high side pressures regularly reaching 300psi on R134A systems. This is a multi-pronged problem:
1) The increased pressure puts an added stress on all of the seals/o-rings that are already potentially 25 years old anyway, and are now in certain places in the system seeing double the pressure they were expected to endure.
2) The pipe work for the condenser and evaporator are not designed to support this pressure. In order to lessen the load on the compressor, R134A systems use smaller diameter pipes in the cooling units in order to increase pressure (if the volume having to be filled is smaller, pressure rises quicker of course). If you compare a 1980s-1990s condenser to a late 90s and up condenser you will see the piping inside is literally about 1/4 the size in most cases. Having the larger diameter piping makes it even more difficult to reach operating pressure. As if it were not hard enough to get the high pressure rated refrigerant up to its optimum pressure range, the evaporator and condenser piping is sized in the opposite direction of what would be beneficial, making it even harder to reach operating pressure.
In addition to these, there is a subset of people that will argue the hose used in R12 systems is too porous for the thinner R134A, however I don't generally subscribe to that. At this age (And even ten years ago) the odds of the hoses already being porous enough to leak at slow amounts are pretty high, and blaming it on the gas type isn't really fair.
The bot***** *****ne here is that if you have NO way of charging it with R12, then R134A is a valid option... just do not expect the world from it. On average R134A systems work at about 70% of the efficiency of R12 systems, and unlike R12 systems you can not just add more gas to improve it... R134A systems have a very narrow range of operation and will quickly not work at all if you exit that range. You can add 10-15% over the specified R12 charge in order to get it closer to its comfort zone, but beyond that you can't bump it up to improve it or anything.
If you have the option, the best thing to do would be to drain the compressor, flush the system out to get all the oil out, and the oil/charge it. If they were available I would recommend putting R134A condenser and evaporators in (there were companies that made these for retrofitting to correct the pressure problems), however to the best of my knowledge these are not available now for this model.