Is this correct
Having laid on the riverbank all day, the young vacationers missed their meals.
No.
http://grammar.ccc.commnet.edu/grammar/notorious/lay.htm
is it lain
yes
Yes, the sentence you provided is grammatically correct. It expresses that the young vacationers spent the entire day on the riverbank and, as a result, they missed their meals.
The sentence is constructed in the past perfect tense with the verb "having laid" (past participle of "lay") to indicate an action that happened before another action in the past. In this case, "laid on the riverbank" is the action that happened first, and "missed their meals" is the action that followed.