By tournament:

1937 Savanna-la-Mar [24 May]
1960 Red Stripe [12 October]

The list may be subject to change if more rugby union sevens tournaments are discovered.

Savanna-la-Mar

The Kingston Gleaner of 29 May 1937 writes of a rugby union Seven-a-side match that took place on Jamaica Empire Day morning. (24 May 1937)

The match took place at Savanna-la-Mar and it was between a Kingston side versus a Country side. The match was over 6 chukkas – a chukka, the time division used in the sport of Polo, lasts around 7 minutes – and Kingston beat the Country side.

This was noted as the first game of the season; and after the Sevens match both sides got together and formed the Jamaica Rugby Football Club.