On the table I have an Oath of Druids and a Forbidden Orchard. My opponent has a Spawning Pit. Assuming there are no other cards/effects to take into account, will I be able to Oath?
Nope.
Tapping a land can't be responded to by my opponent, but is the giving of the token a triggered effect that will give him priority?
Forbidden Orchard's token generation is a triggered ability. After it resolves, your opponent will have priority at least once and be able to sac the token before you can get to a point where Oath would trigger.
And Oath's oracle text defines it as the beginning of my upkeep--so presumably I can't give him a token during my upkeep since Oath has already failed to trigger, right?
Correct. He'd have to have the token at the beginning of your turn, through the untap step into the upkeep, and only then would Oath see it there and trigger. Once players have priority in the upkeep step, it's too late for beginning-of-upkeep triggers to occur.
Can I give a token at the very beginning of my upkeep triggering Oath or is that too late?
See above, it is too late.
Is there some way I can get around all of this, or is Spawning Pit just a really good Oath-hate card?
No way that I can see. Note, too, that regardless of Forbidden Orchard giving your opponent priority, the Oath trigger also uses the stack. If you did manage to get your opponent to have a creature for the beginning of your upkeep and let Oath trigger, he could sac the creature with the ability still on the stack, and it would fail to resolve when it re-checked the condition.
