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The Paladin's Lay on Hands feature says:

As an action, you can touch a creature and [...] restore a number of hit points to that creature.

It also goes on to say:

Alternatively, you can expend 5 hit points from your pool of healing to cure the target of one disease or neutralize one poison affecting it. You can cure multiple diseases and neutralize multiple poisons with a single use of Lay on Hands [...]

Could I opt to do both at the same time? Perhaps an ally was struck by a crossbow bolt with drow poison, and I needed to both heal them as well as remove the poison in one go; is that permissible?

As written, it seems vague. The first paragraph specifies the use of an action to heal hit points, but the second simply indicates an alternative way for the hit point pool to be expended, but doesn't seem to indicate that the use is exclusive from the first.

Unfortunately not.

RAW, the use of Lay on Hands is heal HP, or cure disease. The keyword in the phrase is "alternatively" (emphasis mine).

Alternatively, you can expend 5 hit points from your pool of healing to cure the target of one disease or neutralize one poison affecting it.

The idea is that in combat, the process of using Lay on Hands only allows for you to either "heal" (HP), or focus more specifically on "curing" (disease), and this process is a "6 second" process.

Outside of combat however, this is less of an issue, so long as you manage your Healing pool points correctly.

No, not in the ordinary sense of "alternatively"

Definition of alternatively:

alternatively: as an alternative, instead, or else -- (in place of, or as an alternative to; "Felix became a herpetologist instead"; "alternatively we could buy a used car")

Thus I do not believe it is really vague. If you neutralize a poison or cure a disease, you do so instead of restoring hit points.