1995 Volkswagen Golf grinding noise while gassing

Noises problem
1995 Volkswagen Golf 4 cyl Manual 146000 miles

My car is intermittently making a grinding or vibrating noise whenever I hit the gas. It doesn't do it all the time, but when it does do it, it's never while idling, turning, or breaking. At first I thought it was the air conditioner acting up (I have a slow leak in it somewhere that my mechanic and I are trying to find still), but it does it even when the AC is off.

Today, I drove about half an hour to an appointment with no weird noises. The appointment took 20 minutes, during which my car was parked in a very sunny lot (it's black), so I'm sure it didn't cool off completely. When I started the car, it was fine. When I gassed it in reverse, it was fine. Then when I started gassing it in 1st gear, I heard a grinding noise. Same for when I started gassing in 2nd gear. As long as I'm actively attempting to accelerate the car, in whatever gear I'm in (foot pushing on the pedal, normal appropriate amounts, nothing crazy), it makes this noise. Halfway home, about 15 minutes into the drive, it stopped making the noise when I would accelerate.

Should I be concerned?

Also, I don't drive this car very much. Maybe 2000 miles a year total, so according to mileage, I'm up to date on oil changes (had one several months ago). Could this be the problem? I'm really hoping it is, because I can't afford an expensive repair right now.