Carson said he raised a $1 million in the 24 hours following last week's debate.
The FEC opinion allows members of Congress to make transfers overseas.
The fundraising site ActBlue said it had "never seen" such a fast response.
In a fundraising email blasted out by the DNC, Obama takes Republicans to task.
However, Sanders' campaign has asked the PAC to cease and desist.
Supporters are encouraged to pitch in at least $3 to win the all-expenses-paid trip.
The DSCC reported raising $3.77 million, compared to $3.3 million for NRSC.
"My brother will be a tremendous President," he wrote in a fundraising letter.
The conservative group is accepting donations for Bush, Walker, Cruz, Paul and Rubio.
The End Citizens United PAC has endorsed 11 Democrats so far.