No. You cannot e-file with form 4582 per the IRS.
Authorized IRS e-file providers are prohibited from submitting electronic returns to the IRS prior to the receipt of all Forms W-2, W-2G, and 1099-R from the taxpayer.
If the taxpayer is unable to secure and provide a correct Form W-2, W-2G, or 1099-R, the return may be electronically filed after Form 4852, Substitute for Form W-2, Wage and Tax Statement, or Form 1099-R, Distributions From Pensions, Annuities, Retirement or Profit-Sharing Plans, IRAs, Insurance Contracts, etc. is completed in accordance with the use of that form. This is the only time information from Pay stubs or Leave and Earning Statements (LES) is allowed.