You say you're getting "continuity on one" of the mustard-coloured capacitors? You mean continuity from one side to the other? In that case it's faulty. Capacitors should measure open circuit (except for larger electrolytics, which take a while to charge up, but eventually should read open circuit as well). If you're getting continuity from one end of the capacitor to the other, it's faulty.
You can get a replacement from any electronic component store. "102" is code for 1 nF (one nanofarad). It's a multi-layer ceramic capacitor, but any type of ceramic capacitor will do. In fact you could also use 10 nF or 0.1 uF (point one microfarad); the value is not critical either.
But this doesn't fit with your earlier description of having 0V on the input of the 7805.
I'm really not sure what's happening, and I'm not confident you're doing everything properly. This is what I think you should do.
Solder-suck the positive hole for the left hand electrolytic. (All left-right references relate to the photo in your first post.) Have a good look and try to see whether there is any metallic collar inside the hole, forming a circuit between the track on the top side and the track on the underside. If so, great. If not, do the fix I described earlier.
If the 7805 is not fitted, do the same with the rightmost hole of the 7805. It also needs to have through-hole plating, to connect to the track on the top side. If the 7805 is fitted, don't worry.
Reinstall the two big electrolytics. Don't worry about reinstalling the relay or any ICs. If the 7805 is on the board, leave it there. If it isn't, leave it off.
Connect your 12VAC to the board. Connect your black meter probe to pin 1 of the 555, and with the red meter lead, measure the voltage on the right hand pin of the 7805. If the 7805 is in circuit, measure the leg. Otherwise, measure the pad.
Let me know what you found out about the through-hole plating, and what voltage you measure on the right hand pin of the 7805.
Everything else you've described is fine. Just do that step and let me know, OK?