AP has two copies of Ichorid and a black creature in their graveyard.
On AP's upkeep, both Ichorids trigger. He resolves the first trigger by pitching the second Ichorid to the first one in order to reanimate the first one.
Then the second now-exiled Ichorid trigger gets to resolve.
Everyone agrees on the fact that the second Ichorid can't be brought back to the battlefield from exile, but can AP, for whatever reason they see fit, decide to pay the cost of the triggered ability and exile the black creature from their graveyard - just for the sake of it?

No, they cannot.

Ichorid's triggered ability has what's known as an intervening-if clause, which is an extra condition written in between the initial trigger condition and the rest of the ability. (It ‘intervenes’ between them.) This condition must be true in order for the ability to trigger in the first place, but it also needs to be true when the ability begins to resolve–if it isn't, the ability does nothing.

It's often helpful to think of an intervening-if clause as being both part of the ability's trigger and as part of its effect. Parsing Ichorid's ability this way, we see that its trigger condition, which it needs in order to trigger, is…

At the beginning of your upkeep, if Ichorid is in your graveyard,…

While the effect of its ability, which happens when it resolves, is:

…if Ichorid is in your graveyard, you may exile a black creature card other than Ichorid from your graveyard. If you do, return Ichorid to the battlefield.

So when the trigger resolves, the player doesn't get the opportunity to exile anything from their graveyard, because they don't meet the necessary condition–Ichorid isn't in their graveyard any more.

Only intervening-if clauses behave this way. Conditional modifiers that specify something happens only under certain conditions can appear in other places in a triggered ability, but it's only an intervening if if it's between the ability's trigger and its effects. If a conditional appears later in the ability, such as on a card like Chronicler of Heroes, it's only considered a part of the ability's effects and is only checked as the ability resolves.