Relay Part Numbers, Read Below:
Hello,
I had the same problem till half an hour ago, and was able to fix my '99 XLT's washer pump for the front thanks to the info on this here best and most useful site I've seen for these trucks. Here's some info for anyone looking to acquire the relays BEFORE getting all cut trying to access them to look up the part numbers.
Basically, there are six (6) relays in the compartment at the bottom of the air filter housing (and just the top of the filter housing [a.k.a. airbox] has to be removed to access/replace relays by the way, not the whole thing as some described). The part numbers, as printed on the relays, are:
F57B-14B192-AA (4 smaller relays)
F0AB-14B192-AA (2 bigger relays)
I pulled all of them out, cleaned all connectors, slightly bent some pins and covered them with spark-plug petroleum-like gelly for better conductivity, but it looks like I will have to dive in there again to change one, because now that my front/rear washer pump is working as good as before, I lost my fog lamps (not that I use them, but I like everything to work). Therefore, a piece of advice: buy at least one of the small relays and replace the ******* right away!
I hope this helps someone to find the proper relays for cheap before starting the procedure, and not racing to Ford after gutting their X and paying $20-40 for a cheap piece of plastic Ford can't even make right (now that I know what to look for, I located a few of these used for $1 each - as long as they work, it's all good for me)

Good luck!