Can Jenny Barnes' enable someone to use acquire assets a second time?
No. This is similar to Charlie Kane's special action. The investigator must not already have performed the same action during the current round.

From the official errata:


  Can an investigator’s “additional” action be used to perform an
  action he has already performed in that round?
  
  No. An investigator may perform an action only once
  each round, even if he is able to perform an additional action,
  such as from the Ruby of R’lyeh Artifact or Charlie Kane’s
  action ability.


There is no reason to suppose that Jenny's ability overrules this limitation.

For reference, here is the actual text of Jenny's special:


  Action: You or another investigator on a City space may perform an Acquire Assets action and gain +2 Influence during that action. 


However, if Jenny uses her special action to allow another investigator to perform an acquire asset action (and gaining the +2 bonus to influence), she can, herself, still perform the acquire asset action normally (no +2 influence bonus).