You cannot claim 501(c)(3) status when setting up an account for a private fundraiser to benefit an individual like this. All donations to this are considered to be
gifts by the donors, not charitable contributions. As such, checks should be made out to the indivdual herself. Alternatively, you can set up a corporation using the title you suggested,and apply for a tax identification number from the IRS, but that requires time, expense, and expertise you may not have, plus filing of tax returns by the corporation. One suggestion: there are web-based services to manage individual fundraisers like this - you may want to look into
GiveForward: The #1 Way to Raise Money for a Loved One , for example. I can't vouch for this service, as my only experience with it is having made a gift myself to a fund raiser for an individual. One advantage is it allows gifts to be made by credit card.
Regarding taxes: the person receiving these gifts has no tax obligation whatsoever. The donors making gifts also have no tax obligations unless they exceed the annual gift tax exclusion amount of $14,000.