Can I print the Republican Elephant logo?

I'm not 100% sure its trademarked or not but have seen it used all over the place being printed on all kinds of items. Since your printing them for a young republicans group, I can't imagine why they would have a problem with that but you never know.

The logo I believe is public domain, and your use sounds legitimate (not a smear campaign/etc). That being said, anyone can sue anyone for anything. Believe the republicans were going to sue cafe press over this issue, but then backed down. Whether or not they would've won, who knows,but it would've probably been damning themselves if they continued to go through.

The logo is trademarked (Federal Trademark Registration 1908397).

But according to these articles, it seems like you can create your own version of the elephant logo. But if you use the exact registered logo, they may take legal action.

RNC fights use of elephant logo - Ben Smith - POLITICO.com
GOP 'Caving' on Trademark Lawsuit Threats | Threat Level | Wired.com

If your client has some official affiliation, maybe they can provide written authorization to use the logo.

If they are a recognized club by the party, they should be able to get permission and the logo for you. Get it in writing that it is ok. Do not be surprised if the answer is no.

The best idea is to try to get the club to give you permission in writing to use the appropriate logos (especially if they are providing you digital logos to print). Failing that, you can talk to your client and simply explain you will slightly stylize it (work with them for approval of what looks cool/good) and that should sufficiently release you from any potential legal claims (ie. that's your workaround of permission to use the existing logo).

Since its a club they probably need permission. If someone was running for office written permission is not needed. I just did some signs for a candidate and after calling the local office that was their response.