If condition cards run out, do investigators not gain the condition?
If condition cards run out, investigators do not gain the condition.


  An investigator can gain a card or token only if it is available.


Source: Reference Guide, "Component Limitations" (page 3)

The Epidemic Prelude's "gains 1 Illness Condition and 1 Clue" is an identical situation as "recover 1 Health and 1 Sanity" or "discard 1 Item and 1 Trinket".
These loses/gains aren't dependent on each other, those are just lists.
If you can't do everything from the list you do as much as you can.
Not recovering health doesn't prevent you from recovering sanity.
Not losing a Trinket doesn't prevent you from losing the item.
Not gaining Illness doesn't prevent you from gaining a clue.