If you want to claim your daughter on your 2020 tax return you will have to wait until her SSN is issued, so filing an extension is a good idea.
The IRS will determine eligibility for the stimulus based on your last tax return (either 2019 or 2020) and will likely send your payment to the bank account where your tax refund was deposited. The stimulus checks will probably go out before your 2020 return is filed, so your daughter will not be included in the stimulus payment. However, I suspect there will be some provision like the Recovery Rebate Credit for people who did not get the total stimulus payment they were due. But that is not a certainty.
For more information please see this TurboTax Help article: How will the third stimulus impact me?
