Yes. An LLC is a legal entity. A legal entity can lease property from others, including from its owners, which of course would include a member of the LLC. Rental payments from the LLC can be paid but you should also review with both your tax advisor and insurance advisor to avoid unexpected matters in those area, such as tax implications/requirements, and to assure insuance coverage for both you and the LLC. A lease where the LLC agrees to hold you hrmless and a pledage of LLC assets as securtiy should be considered to provide priority to you over other potential business creditors.
A frequent issue with LLC or partnership dealings with members/partners has to do with favorable self-dealing transactions. In your case, the van may only be leased at a commercially reasonable terms. Another problem ocurs when the member/shareholder continues to use the property for his/her personal use. Any of these types of transactions may be construed as a distribution of income, or a return of capital to the member who is receiving payments from the LLC.
You will also want to review the LLC/partnership agreement to make sure that the entity is permitted to enter into transactions with its members.