An animator searches his own body to extract the memories, emotions and anxieties that (will) enrich his work.

I’ve put "will" in parentheses because it’s not clear from the context whether the enriching will take place in the future or if the French future tense actually refers here to the present, as it sometimes does.

I believe the sense of venir in this case is the following (from the TLFI online dictionary):

• 3. [Indique une mise en relief de l’action exprimée par l’inf.] Vous venez prétendre que; ne venez pas dire que. Le souvenir vague et confus venait étinceler par moments à son esprit (SUE, Atar-Gull, 1831, p. 18). Mais une occupation vint la distraire (FLAUB., Cœur simple, 1877, p. 25).

So venir doesn’t have a specific meaning here. It merely emphasizes the action expressed by the following verb. That’s why I left venir untranslated above.

An explanation seems plausible; it is found in the TLFi.

III. − Empl. semi-auxil.
A. − Venir + inf.
2. [Indique le caractère fortuit ou possible de l’action exprimée par l’inf.]
♦ On viendra sans doute dire que.
♦ Que le courtisan ne vienne pas ramper sur mon amour; que la vipère ne vienne pas jeter son venin sur mes fleurs. (Dumas père, Intrigue et amour, 1847, 7etabl., 3, p. 274)

There is not necessarily a well established reason why a particular emotion or fear will be the basis for a contribution to this author’s work, and this construction is meant to express this idea.

The translation could then be as follows.

• emotions and anxieties that will happen to feed his work.