This could be answered in many ways that all depend upon who is being lobbied and what their position and organization to which they belong is. (e.g. Senator, Chairman, Legislator, etc.)
Some suggestions that are irrespective of who or what is being lobbied:
target : a place, thing, or person at which an attack is aimed
Has the connotation of the lobbying being an attack or some kind, that the end goal of the lobbying is somehow harmful to the person using this term to describe the action. Synonyms with varying degrees of this connotation include: prey, quarry, game
mark : 15.b. slang the intended victim of a swindler, hustler, or the like: The cardsharps picked their marks from among the tourists on the cruise ship.
Connotes a degree of deceit on the part of the lobbyist. Combined with easy mark, connotes one who is easily taken advantage of. Synonyms include: dupe, (clay) pigeon, sucker, doormat.
More neutrally, we have the logical lobbyee (or lobbee) and lobbied as others have suggested, but you could consider:
colleague : an associate in a profession or in a civil or ecclesiastical office
Has no connotation of what the association with the lobbyist is. Combined with a adjective phrase, such as colleagues in the House and Senate, it could fit into your usage. However, it could cast the impression that lobbyists and private organizations are an essential part of the system. Synonyms include: counterpart, peer, confederate. Near synonyms with a definite negative connotation include: crony, chum, co-conspirator, accomplice.
Neutral, near synonyms to lobby include influence, solicit, approach, accost, which have more reasonable forms of past-tense-direct-object: the influencee/d, solicited, approached, or accostee.
You could also use a general term which makes their role obvious when partnered with lobbyist, such as the power holder, incumbent, or office holder.