2 questions.

First question:

Are you a tail?

Now we will figure out who to ask the next question to:

The answer to Q1 is "yes". Two of the deterministic tails will slap and two will crack. The random tail will join one of those groups. Simply choose the two tails with the same response. They are guaranteed to not have a random tail in their group.

Now ask this group:

"Is laundry powder in stock?"

Explanation:

If you get two different answers, the true answer is "no" because the tail in your group which changes answer based on truthiness has just done so. (One tail in your group will give the same answer no matter what, but they don't matter right now). Using similar logic, if you get two of the same answer then the true answer is "yes"

New answer: 1 question.

If I were to ask you if there was laundry powder in stock, would you answer with a crack?

Explanation:

If there is laundry powder, tail A answers "crack" to "is there laundry powder?" and "crack" to "would your answer be a crack?"; tail B answers "slap" and "crack". Thus either way, there are at least 3 answers of "crack" (counting tail D, not caring about the random). So if there are >= 3 cracks, the answer is yes. With similar logic >= 3 snaps means no.

Outdated answer, for posterity

2 questions.

First question:

Are you a tail?

Now we will figure out who to ask the next question to:

The answer to Q1 is "yes". Two of the deterministic tails will slap and two will crack. The random tail will join one of those groups. Simply choose the two tails with the same response. They are guaranteed to not have a random tail in their group.

Now ask this group:

"Is laundry powder in stock?"

Explanation:

If you get two different answers, the true answer is "no" because the tail in your group which changes answer based on truthiness has just done so. (One tail in your group will give the same answer no matter what, but they don't matter right now). Using similar logic, if you get two of the same answer then the true answer is "yes"