Most likely the psu doesn't have a pci-e plug either, so expect to lose 2 4 pin connections.
If it was a system made to handle SLI 7800's it has two PCI-E connectors.
And SLI 7800's take about 150 watts.
He should be good to run a modern single GPU that isn't extremely demanding.
5770 would run just fine.
Confirmed the 7800's used a 6 pin connector.
My guess is that since it's OEM but it could be customized with 7800's in SLI...
It probably has two 4 pin + 2 connectors or a single 4 pin + 2 connector and a 6 pin connector.
If OP would open up his case and look at the connectors he could just tell us.
Either way I'd bet $100 OP's PSU can handle a 5770