- Does anyone know who/whom I can speak with about that?
As noted, the question to ask is what function the Wh-word has in the subordinate clause. If it's the subject of that clause, it has to be who. But I is the subject, so that's not a consideration.
If it's not the subject, then what is it? It would appear that the Wh-word refers to the object of speak with:
- I can speak with
Indef about that
So, since it's an object, you are officially allowed to use whom, if you really want to.
Remember, however, that using whom at all, ever, marks your speech as more formal than usual.
Remember also that most other English speakers don't know the official rule (Consider: you're confused about it -- how many others are?), and they often make up their own erroneous rules, which will cause them to judge your use of whom as incorrect, even if it is officially correct. So if you're using whom to be "correct", you'll have to be satisfied by your own superiority. In other words, you can't win.
The only reasonable solution is to never use whom. It serves no purpose and fulfills no need that you can't satisfy by using who (or occasionally that) instead. The only place it's actually required in English is as the object of a Pied-Piped preposition, i.e,
- Does anyone know with whom I can speak about that?
is OK, but
- *Does anyone know with who I can speak about that?
Pied-Piping requires whom, but stranding the with at the end is equally grammatical,
since Pied-Piping is optional; and it's far more fluent:
- Does anyone know who I can speak with about that?
... though my conversational American English would prefer:
- Does anybody know who I can talk to about that?