The Illinois Appellate Court has answered the question in the affirmative.  LLC Law Monitor has the details:

The trial court based its lien ruling on Fitzgerald v. Van Buskirk, 306 N.E.2d 76 (Ill. App. Ct. 1974), which had held that a building contractor “acted in the capacity of a joint venturer with the defendants and in such capacity was not a person entitled to a mechanic’s lien.” Id. at 78. The Appellate Court agreed with the Fitzgerald rule, noting that “an owner or co-owner of property may not claim a lien against his or her own property.” Peabody-Waterside, 2013 WL 4736714, at *2 (citing Bonhiver v. State Bank of Clearing, 331 N.E.2d 390, 398 (Ill. App. Ct. 1975)).
But the Appellate Court disagreed with the trial court’s conclusion. The court pointed out that: 

• an Illinois LLC is a legal entity distinct from its members (805 Ill. Comp. Stat. 180/5-1(c));
• an LLC member is not a co-owner of the LLC’s property (805 Ill. Comp. Stat. 180/30-1(a));
• a member of an LLC owns only its membership interest in the LLC; and
• sharing in the profits and losses of an LLC does not make the LLC’s members jointly interested or co-owners of the LLC’s property.

 Id. at *3. The court differentiated LLCs from joint ventures, which it said are not distinct legal entities. Id.
The court concluded: “Given that [the contractor], as a member of [the LLC], is not jointly interested in the property, nor is it a co-owner of the property, its mechanic’s lien must, therefore, be valid.” Id.
 

The Illinois court's reasoning seems sound and the factors it cites are common to LLC statutes, including the pervasive Delaware LLC Act.[1]  Moreover, it's typical, at least in Delaware, for LLC Agreements to specify that members can conduct business with the LLC provided it's done at an arms-length basis on terms and conditions customary to the industry.[2]