Great question!
The answer is they do it quite a bit. Except during those rare times in which the Great Council is in session, there are no legislative politics per se – governments take action through the decrees and decisions of the king and his ministers, so if you want anything from the government, you have to do it through lobbying.
And we have a lot of examples of this in the series, leaving aside the omnipresent chivvying for lands and titles:
…the septa could not have known that today’s court would be anything but the usual tedious business of hearing petitions, settling disputes between rival holdfasts, and adjudicating the placement of boundary stones. (Eddard XI, AGOT)
Lord Redwyne asked only for thirty years’ remission of the taxes that Littlefinger and his wine factors had levied on certain of the Arbor’s finest vintages. When that was granted, he pronounced himself well satisfied and suggested that they send for a cask of Arbor gold, to toast good King Joffrey and his wise and benevolent Hand. (Tyrion III, ASOS)
Jalabhar Xho was the first to petition her that day, as befit his rank as a prince in exile. Splendid as he looked in his bright feathered cloak, he had only come to beg. Cersei let him make his usual plea for men and arms to help him regain Red Flower Vale…Lord Hallyne of the Guild of Alchemists presented himself, to ask that his pyromancers be allowed to hatch any dragon’s eggs that might turn up upon Dragonstone, now that the isle was safely back in royal hands…(Cersei VIII, AFFC)
Indeed, one could argue that courtiers are essentially lobbyists with better fashion sense.
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