AMP 350206-1 is the socket, but I have not found a source for them. GPE has looked for them as well.

The pins are AMP 350068-1, and GPE carries them: https://www.greatplainselectronics.com/proddetail.asp?prod=350068-1

TE Connectivity owns AMP's stuff, but they likely won't produce any of the stuff marked as obsolete, like the socket. The last time I communicated with TE, they suggested this socket as a replacement, with solder lugs instead of crimped pins:

http://www.te.com/usa-en/product-1419106-1.html
http://www.te.com/commerce/DocumentDelivery/DDEController?Action=showdoc&DocId=Data+Sheet%7FKH_Accessories%7F1115%7Fpdf%7FEnglish%7FENG_DS_KH_Accessories_1115.pdf%7F1419106-1

What are the specs on the relay? From working with the electronic-based allied leisure games, they used a 4PDT relay, like this:
http://www.newark.com/nte-electronics/r12-17d3-24/power-relay-4pdt-24vdc-5a-plug/dp/10M0334
http://www.alliedelec.com/nte-electronics-inc-r12-17d3-24/70011899/

I'm not sure if GPE's is the same or not:
https://www.greatplainselectronics.com/proddetail.asp?prod=LB4H-24DS

AMP_350206-1.pdf

That footprint looks very ordinary.
chk with fry's, or just about any electronics shop.
They may have that for you

Hey, this was in my Allied Leisure Getaway pin. Looks very similar, want a shot a trying it?

image (resized).jpg

image (resized).jpg

image (resized).jpg

I think John's Jukes in Canada has some parts for those machines I would try him first before going to crazy looking for stuff from the "normal sources". He isn't cheap but he does have a lot of oddball stuff no one else has!

yup, thats a standard 4 poll d throw, 24 volt coil, not rare at all.

To bring this thread to a conclusion --

If just replacing a crimp pin on the socket: 350068-1 (Obsolete but I have remaining stock).

If replacing entire shooting match:
Modern replacement socket is 1419106-1 also known by older part number 27E006 .
Modern relay is LB4H-24DP (Rayex) or MY4-DC24 (Omron).
These relays do not include the hold down screw that the original had.
To get a relay with a hold down screw -> Potter & Brumfield KHAU-17D11-24 (new part number = 4-1393122-4).

Many of these can be found at Great Plains Electronics.

There's a myriad of obsolete part numbers for other replacement relays such as the Fujitsu FRL-263 listed above, not worth the effort of listing the obsolete parts.