It can be done, but there are some considerations.
First, if you want to be able to do APRS and also do voice on the same radio at the same time, the rig you have isn't going to work. But, if you just want to do APRS on the national VHF APRS frequency, 144.390, you're set. I strongly recommend the Kenwood TMV71A or TMD710 if you want to do both things at once - you can assign the APRS to one VFO and voice to the other. The Yaesu rigs don't work like this - the digital follows the active VFO.
If you have a laptop that will work, you can avoid the TNC, and might even be able to use your existing GPS. It might be a very good idea to look for an older laptop that you can beat up without any tears - you can find them pretty cheap. I found that older Lenovo/IBM Thinkpads will run fine on the 13 volts available in a car. You may be able to use your existing GPS if it will output NMEA-compatible data. Most GPSs don't have serial ports anymore, they only have USB, and many of those that have USB aren't set up to send the NMEA data, so you will need to look into that. If the GPS will send NMEA data via USB, you can use it with a laptop running an APRS program. Some of the APRS programs out there will support the AGWPE packet engine program, which emulates a TNC in a PC. Then, you just need to construct a soundcard interface to the radio, or buy one.
Now, I've done exactly what I describe here, using a RASCAL soundcard interface and an old IBM Thinkpad and a handheld Garmin GPS. It all worked, most of the time, though the laptop was a little too slow to keep the software happy. I ran UI-View and AGWPE - along with a mapping program - quite a load for an old laptop, but more recent ones should handle it OK. It took a lot of fiddling and advice from others to get it running. I couldn't use it in motion and I suspect that having the laptop running on the seat next to me would get a ticket in a lot of places.
Anyway, the first time I saw an old Kenwood TMD700 for sale cheap, I pounced on it, and I haven't regretted it at all. I've since upgraded to a TMD710, and I like it much better. But, the learning exercise of getting it to work a different way was probably worth the effort. It will certainly not be 'plug and play', though. Even the Kenwoods require a bit of setup to get everything working, and they both require a GPS with a serial NMEA output. On the D710, I use the neat little GPS module from GreenLight Labs. I used the old Garmin and some small GPS receiver modules with the D700.
The only way to really get all the function out of APRS is to use a computer and APRS software. But the Kenwoods will do almost everything.
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