IANAL and I have zero experience doing anything remotely like this.

But from my cursory reading of the IRS instructions on their website, doing this in the US specifically is more or less a two step process:

1. Form an organization by filing with your local state. Every state handles this process differently; in some (most?) this can be as straightforward as filling out a form, paying a processing fee on the order of $100 or so, and waiting for approval. Just don't form an LLC in particular, as that complicates the next step.

2. Fill out Form 1023-EZ with the IRS. This requires proving your organization qualifies for tax exemption (it is not clear to me whether this would) and a filing fee of $275. Your org also cannot possess more than $250,000 of assets, cannot receive more than $50,000 of revenue from donations within the span of a year, and cannot be registered as an LLC. If you fail to meet these, you need to fill out the regular Form 1023, which I believe is more involved and has a more expensive filing fee.

If both of these forms are accepted, kapow! You are now a tax-exempt organization, and other corporations can charitably donate to your project for tax breaks. Just remember to do your station-keeping tasks like filing your annual company and tax exemption status renewals, reporting your earnings to the IRS, and sending receipts to donors.