Yes, your partner cannot file a correct personal tax return until he has the K-1 from the partnership tax return. He does not have to wait for the IRS to process the return, he can file his personal tax return with the K-1 from the partnership as soon as the partnership files the tax return.
Note that you can print and mail your partnership Form 1065 with your application for an ITIN, according to the instructions for Form W-7, pasted below:
"Your original tax return(s) for which the ITIN is needed. Attach Form W-7 to the front of your tax return. If you’re applying for more than one ITIN for the same tax return (such as for a spouse or dependent(s)), attach all Forms W-7 to the same tax return. Leave the area of the SSN blank on the tax return for each person who is applying for an ITIN. After your Form W-7 has been processed, the IRS will assign an ITIN to the return and process the return.
Click these links for Form W-7 and Instructions for Form W-7.
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