Can I hire myself as a contractor for my own LLC?

There is no tax reason why a multimember LLC could not engage you as a subcontractor.  Your LLC income is reported on a 1065 and K-1, and the subcontractor income would be on a schedule C as part of your personal tax return.

The question should be directed to the other members of the LLC and your (hopefully written) partnership agreement.  You would be shifting part of the income exclusively to yourself.  Is that a conflict of interest?  Is it allowed by the partnership agreement?  Would your partners approve?  If you have a partnership agreement, what did you each bring to the table, and why are these particular skills or tasks not part of your participation under the agreement, why are they something extra?

I don't think the IRS will care as long as you pay your taxes.  But your other partners might have an interest in the matter. 

by the way is this an LLC with an S-Corp election. I ask because you say you're a shareholder. In this case, I would think if the other owners agreed, you could get paid for your services. However, I think it would have to be a w-2 since you would be an employee of the corp.  By the way as an S-corp, all owners who are active in its operations are supposed to receive a salary. 

the downside is that w-2 wages don't qualify as QBI. If paid as an IC, I don't know if the IRS would object. if it does and it's sustained, the S-corp would be subject to substantial penalties