No. The Trainer section on the entry form is really asking who is responsible for the horse while at the show. This individual must be a member of both USEF and USDF.
The definition of a Trainer under USEF General Rules:
GR404 Accountability of Trainers and Other Persons Responsible
4. A trainer is defined as any adult or adults who has or shares the responsibility for the care, training, custody, condition, or performance of a horse and/or pony. Said person must sign the entry blank of any Licensed Competition whether said person be a trainer, owner, rider, agent and/or coach. Where a minor exhibitor has no trainer, then a parent, guardian or agent or representative thereof must sign the entry blank and assume responsibility as trainer. The name of the trainer must be designated as such on the entry blank. It is the responsibility of trainers as well as competition management to see that entry blanks contain all of the required information. The responsibilities of a trainer include, but are not limited to the following:
a. for the condition of a horse or pony at a Licensed Competition (whether or not they have signed an entry blank),
b. to guard each horse and/or pony at, and sufficiently prior to, a Licensed Competition such as to prevent the administration by anyone of, or its exposure to, any prohibited substance, and
c. to know all of the provisions of this Chapter 4 (including any advisories or interpretations published in equestrian) and all other rules and regulations of the Federation and the penalty provisions of said rules. For purposes of this rule, substantial evidence means affirmative evidence of such a clear and definite nature as to establish that said trainer, or any employee or agent of the trainer, was, in fact, not responsible or accountable for the condition of the horse and/or pony. If any trainer is prevented from performing his or her duties, including responsibility for the condition of the horses and/or ponies in his or her care, by illness or other cause, or is absent from any Licensed Competition where horses and/or ponies under his or her care are entered and stabled, he or she must immediately notify the competition secretary and, at the same time, a substitute must be appointed by the trainer and such substitute must place his or her name on the entry blank forthwith. Such substitution does not relieve the regular trainer of his/her responsibility and accountability under this rule; however, the substitute trainer is equally responsible and accountable for the condition of such horses and/or ponies.
5. The trainer and owner acknowledge that the trainer represents the owner regarding horses and/or ponies being trained or managed, entries, scratches for any reason and any act performed on any horse and/or pony under the care and custody of the trainer.