Client letter is not in the expected format for H1B

HI,

This is Prady, I am working as consultant at Caterpillar Inc. and applying for H1B this year.
I have one layer(vendor) in between Caterpillar and my Employer. Caterplillar gave me
client letter saying “they have contract with Vendor and vendor has requested for my H1B and has right
over my hiring, payment etc”. They did not mention my employer name on the letter.

Now my vendor is saying Caterpillar is not willing to change anything on the letter and would
not put my employer name as they do not have direct contract with them. But vendor is ready to give a supporting
letter saying contract between him and my employer.

Please let me know, If I can submit this client letter with supporting Vendor letter for my H1B?

Thanks in advance.