Yes. A trust is a legal entity, so for the purposes of USPS Form 1583 treat it the same way you would a business or organization. If mail will arrive addressed to the trust, the trust needs to be on a 1583.
How to fill it in
Section 3 — Type of Service Requested: select Business/Organization Use. This also covers your personal mail, so you are not giving anything up by choosing it.
Section 4 — Name of Applicant: a natural person (typically you).
Section 7 — Business/Organization Information: the trust goes here. Use the full legal name of the trust in 7a, including the word "Trust". For 7b, Type of Business, a short plain-language description is all that is needed — for example, "revocable living trust".
With that combination, the trust is on file as an addressee and you are on file as the natural person behind it, so mail addressed to the trust can be accepted and processed.
How many forms do I need?
A single 1583 covers one entity plus one individual. So the one form above covers both your trust and you personally — you do not need a second form for your own mail.
You need an additional form for each additional entity. A trust and an LLC is two forms. Two trusts is two forms.
More detail
For a field-by-field walkthrough of the whole form, see What is the purpose of each field on the USPS 1583 Form?