One company is telling me that their taxes are behind a whole year and I may not get my 1099 until June or later. Should I file my taxes now and amend them later?

[fanfare]

Amending is troublesome at best, so that's not a good idea.

request an extension to file instead.

figure your tax as if you had the document. Pay in full by April 18th

If you know the exact amount of a1099-NEC you can enter it now and file.

@ToolPunk 

[Opus 17]

What kind of 1099 are you waiting for? If you are self-employed, you are required to report all of your income and expenses from your own accurate records, regardless of whether your clients or customers give you 1099-NEC forms.  You can go ahead and file without, including the form as long as you include the income. If the IRS sends a letter asking for more information, you can send them a copy of your PNL, or other records from your business showing that you already included the income.  

If you are waiting for something else, like a 1099B or a 1099R, then you should probably wait.