The Chan Zuckerberg Initiative is not a 501(c)(3) tax-exempt entity. It will be treated as a pass-through entity for income tax purposes so that income, expenses and other tax attributes will flow through to the members (the Zuckerbergs) who will report that income on their individual income tax return. There may be little taxable income to pass out to the Zuckerbergs because Facebook stock does not currently pay dividends.
Because it is not a private foundation, the Zuckerbergs will not receive a charitable contribution deduction when they contribute their stock to the LLC. But this may not matter to them because without Facebook paying dividends, their income is most likely not high enough to take advantage of the charitable deduction. If they had made the contribution to a private foundation, the charitable contribution that they could take each year would be limited to 20% of their adjusted gross income. This could be carried over for five years, but they would never be able to utilize the full tax deduction of $45 billion.
If the Chan Zuckerberg Initiative does decide to make contributions to charity, as a pass-through entity, the LLC would pass those deductions out to the Zuckerbergs. Instead of giving cash to the charities, the LLC would most likely give appreciated Facebook stock to the charity which would produce a charitable deduction equal to the fair market value of the stock at the date of the gift. This would not trigger a capital gain recognition at the LLC level, and the Zuckerbergs would therefore avoid paying capital gains tax on the shares gifted.