In Texas, according to
Motor Vehicle Title Manual, Pg 25-10:
"If a motor without an identifying number, such as a Ford motor or a motor from some 1956 and later model vehicle without a number is installed in a vehicle which requires a motor number, an assigned number must be obtained from this department. (Refer to Chapter 13)."
This applies to a vehicle you already have a title to. You have two problems here.
1. You have a vehicle with no title
2. You have a vehicle with no engine number to trace the original title with.
So..from Chapter 13 page 13-12:
"B. If the maufacturer's vehicle identification number has been removed, changed, or obliterated and the true identity of the vehicle cannot be determined by the inspecting officer, a court order or Tax Collector's Hearing Order will be required to establish ownership."
(Note: All bold text is bold in the original document.)
The best thing for you to do is find something, such as a registration receipt or insurance card, showing the original title number. With that number you can try to contact the last registered owner. If you can contact that owner perhaps he will file for a lost title and transfer it to you. Offer to pay all expenses plus a tip. If he won't file for lost title ask him to sign a statement that he has sold the vehicle and has no further interest in it. This will help you at the Tax Collector's Hearing. Take all the documentation you have to the tag office to apply for the hearing.