I am not sure about your state, but this would work in some states.
You could start the project as a sole proprietor, and then once the LLC is formed, you transfer your project to the LLC, which then continues the project. If possible, you then bill the customer after the LLC is formed.
Any payment received before then would need to go on your personal taxes, since the LLC didn't exist yet. You either need to pay taxes on it under your personal taxes, or you can negate out the transaction by hiring your LLC.
For example, let's say you accepted a deposit of $1000. That $1000 is your income on your personal tax return as business income. But, the IRS allows you to deduct expenses. So you personally hire your LLC to complete the project and you pay the LLC $1000. Your personal profit is now $0, and you pay no personal tax on it. But now, your LLC has $1000 of income that it has to pay taxes on. You basically shift the tax burden to the LLC. This is acceptable since the government still gets its taxes.
If you are going to do this, talk to your accountant to make sure that you do this the proper way.
But take note that before the LLC is formed, any agreements listing the LLC as a party could be deemed be invalid. You could word the contract or agreements so that you are allowed to transfer the project to the LLC. You probably could get away with the contract even saying the project will be completed by your LLC. What you can't do is sign the contract as your LLC. You would have to sign the contract personally. This means no limited liability while the project is in your possession, so be careful.
To protect yourself, you may want to consult an attorney concerning the proper wording of a contract.
So it's possible to start work before the LLC is formed, but there is more paperwork involved.