The Collaborative International Dictionary of English v.0.48:

Solicit \So*lic"it\, v. t. [imp. & p. p. Solicited; p. pr. &
   vb. n. Soliciting.] [F. sollicier, L. sollicitare,
   solicitare, -atum, fr. sollicitus wholly (i. e., violently)
   moved; sollus whole + citus, p. p. of ciere to move, excite.
   See Solemn, Cite.]
   1. To ask from with earnestness; to make petition to; to
      apply to for obtaining something; as, to solicit person
      for alms.
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            Did I solicit thee
            From darkness to promote me?          --Milton.
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   2. To endeavor to obtain; to seek; to plead for; as, to
      solicit an office; to solicit a favor.
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            I view my crime, but kindle at the view,
            Repent old pleasures, and solicit new. --Pope.
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   3. To awake or excite to action; to rouse desire in; to
      summon; to appeal to; to invite.
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            That fruit . . . solicited her longing eye.
                                                  --Milton.
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            Sounds and some tangible qualities solicit their
            proper senses, and force an entrance to the mind.
                                                  --Locke.
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   4. To urge the claims of; to plead; to act as solicitor for
      or with reference to. [Obs.]
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            Should
            My brother henceforth study to forget
            The vow that he hath made thee, I would ever
            Solicit thy deserts.                  --Ford.
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   5. To disturb; to disquiet; -- a Latinism rarely used.
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            Hath any ill solicited thine ears?    --Chapman.
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            But anxious fears solicit my weak breast. --Dryden.
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   Syn: Syn. To beseech; ask; request; crave; supplicate;
        entreat; beg; implore; importune. See Beseech.
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WordNet (r) 3.0 (2006):

solicit
    v 1: make a solicitation or entreaty for something; request
         urgently or persistently; "Henry IV solicited the Pope for
         a divorce"; "My neighbor keeps soliciting money for
         different charities" [syn: solicit, beg, tap]
    2: make amorous advances towards; "John is courting Mary" [syn:
       woo, court, romance, solicit]
    3: approach with an offer of sexual favors; "he was solicited by
       a prostitute"; "The young man was caught soliciting in the
       park" [syn: hook, solicit, accost]
    4: incite, move, or persuade to some act of lawlessness or
       insubordination; "He was accused of soliciting his colleagues
       to destroy the documents"
    5: make a solicitation or petition for something desired; "She
       is too shy to solicit"

Moby Thesaurus II by Grady Ward, 1.0:

55 Moby Thesaurus words for "solicit":
   accost, address, angle for, appeal to, apply, approach, ask,
   ask for, beg, beseech, bespeak, bid for, call, call on, call upon,
   canvass, challenge, claim, court, crave, demand, desire, drum,
   entice, entreat, exact, fish for, go, hustle, implore, importune,
   look for, lure, make a pass, make advances, make an overture,
   overture, pander to, petition, pop the question, postulate, pray,
   proposition, refer, request, require, requisition, resort, seek,
   sue, sue for, supplicate, throw a pass, turn, woo