Answer

Yes, a player can take the Camp action even if they have no cards left in their hand or discard pile, and they still gain one Health.

Explanation

• The Camp action is described as follows:
  "When you take the camp action, immediately shuffle all (play area, discard, deck and hand) of your cards back into your deck and gain one . You DO NOT return to San Andreas when your deck is reshuffled this way. This ends your action phase." (Page 22)

• The rule does not specify any requirement to have cards in your hand or discard pile to take the Camp action. It simply instructs you to shuffle all your cards from all zones back into your deck (even if some or all of those zones are empty) and then gain one Health.

• Therefore, even if your hand and discard pile are empty, you can still take the Camp action and gain one Health.

Source: page 22.

Source: Core Rulebook (pp. 13, 19, 22, 23, 7)