It sounds as if the 8 relays together take more than 1A. Unfortunately, that ebay page doesn't say how much current the relays take, and the picture of the board does not have enough resolution for me to read the part numbers. There is no need to use a regulated supply for powering relays, so I would ditch the 5v relay board and the 7805, and buy the 12v version of the relay board instead. The 12v relays should take less than half the current of the 5v versions, but ask the supplier how much current they take, just to be sure.
The other possibility is that the 8 relays do take no more than 1A between them, but you don't have a heatsink on the 7805 so it is going into thermal shutdown. At 1A and 7v voltage drop, the 7805 will dissipate 7W, so it definitely needs a good heatsink.