"on-the-fly" installing of new firmware through (usually a serial) connector is known as In System Programming, ISP. A lot of modern chips have such a feature incorporated, so you don't need to design in any extra hardware to have such a feature. ISP is usually associated with "flash memory", a type of PROM which is electrically erasable.

If you want to design a device which offers ISP functionality without using chips which have this feature built in, you will need to incorporate circuitry which is able to erase and write the program memory chip, generally under the CPU's control which, for the occasion, will be running from a fixed, different program memory chip usually referred to as the boot ROM, and especially in case of 8051 based designs, some scheme to temporarily map the program memory chip as external RAM (there is no instruction to write to program memory!)

Now you are talking about EPROMs. That sounds like it will be impossible to reprogram your existing devices in an ISP fashion. First of all, the board would have to be able to erase the EPROM chips, meaning it will NEED a UV lamp inside the enclosure. Highly unusual... If your device doesn't have this: no chance. Then your device needs to have a separate boot ROM which can not be erased while the normal EPROMS get UV blasted. If your device doesn't have a special, second, unerasable EPROM, again: no chance. Next, you will need an address decoding glue which allows to make the EPROMS appear as external RAM chips to your 8051, otherwise you can't write to them. If your glue doesn't allow for that: no chance. Finally, your device will have to have the ability to provide a programming voltage to the Vpp pin of the EPROMS. If your device doesn't have that: no chance. So. If it is about EPROMS, you see your device will have to incorporate a lot of special and exotic hardware in order to make any on-the-fly reprogramming possible, which I doubt is the case in your design. (I've never seen an ISP capable, EPROM based device at all)

Another option which is open to you (as Joseph suggested) is to make the step towards ISP capable chips and replace the non-ISP capable chips in your existing device with them. You will need a pin-for-pin compatible replacement, and the ISP capability will relate to the own chip's internal program memory ONLY. I don't know what your hardware looks like exactly, but if it is an 8051 type chip running from an external EPROM, from the time you replace the MPU by an ISP capable chip, the external EPROM will no longer have a role. You will have to run from the chip's internal flash from that moment on to have any benefit of the ISP capability.