Usually, no, this should not affect your refund. The Social Security Administration (SSA) shares the name change with the IRS. The only time it would affect your refund, is if you did the name change within the last week or so and the IRS has not received your new name from the SSA.
If your income tax return(s) got "Accepted" by the IRS, that means the IRS has your new name. Sometimes, the IRS, for security reasons, may send you a letter to verify who you are. Just follow the instructions on the letter. See Taxpayers Receiving Identity Verification Letter Should Use ... - IRS.gov
Related information: Does it matter whose name goes first on a joint return?
