It appears that trailers under 7,000 pounds don't have to be inspected in Texas. In order to register a used trailer less than or equal to 4,000 lbs. gross weight, you need the Bill of Sale + Registration Receipt.

https://www.txdmv.gov/motorists/buying-or-selling-a-vehicle/trailers

If you were to get an inspection, this is what they inspect. But your trailer shouldn't require this.

https://www.dps.texas.gov/rsd/vi/inspection/inspectioncriteria.aspx

What's Inspected by Vehicle Type

Items of Inspection for a Trailers or Mobile Homes

Inspect All Trailers, Semitrailers, Pole Trailers, or Mobile Homes Exceeding 7,500 Pounds Actual Gross Weight or Registered Weight For: (Listed in suggested order of inspection) Refer to Federal Motor Carrier Safety Regulations, if required.

* Check for evidence of Financial Responsibility on towing vehicle

1. Brakes (system) (If gross weight exceeds 7,500 pounds)
2. Tires
3. Wheel Assembly
4. Safety Guards or Flaps (if four tires or more on rearmost axle) Pole trailers exempt.
5. Tail Lamps (2)
6. Stop Lamps (2)
7. License Plate Lamp (1)
8. Rear Red Reflectors (2)
9. Turn Signal Lamps
10. Clearance Lamps
11. Side Marker Lamps
12. Side Reflectors
13. Side Marker Lamps and Reflectors (30 feet or more in overall length)
(Refer to Reference Section as per lighting diagrams and as applicable to the particular trailer being inspected.)
14. Serial or Vehicle Identification Number
15. Window Tint.