For the noun - "extracting gifts." Someone who's extracting gifts doesn't necessarily explicitly solicit them (although it's one method), but often does so implicitly by the nature of the relationship entered into. Some examples:
https://www.google.com/#q="of extracting gifts"


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Note added at 4 hrs (2016-08-18 19:51:50 GMT)
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From the above search, since evidently some are unfamiliar with this usage of "extracting gifts" to describe a particular sort of malfeasance (which can include outright solicitation thereof, but doesn't have to; the request for gifts may be tacit and implicit).

More broadly, she accuses Banier of extracting gifts from her mother at moments when she is physically at her weakest.
http://www.france24.com/en/20100701-bettencourt-trial-charac...

The second Army drill instructor to face charges of extracting gifts from trainees at Ft. Leonard Wood was found guilty today on two of the three charges against him.
https://www.newspapers.com/newspage/2432120/

(Lovely one about Chaucer's Wife of Bath)
... and in lines 407-50 she forthrightly admits her mercenary motives as she describes her tactic of extracting gifts from a husband before she will allow sex and divulges her methods of scolding or wheedling her husbands to induce submission to her.
http://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=17084&...