Can I equip an Equipment on my unit with Relentless Pursuit?

Yes, you can attach an Equipment to your unit using Relentless Pursuit.

According to the card text for Relentless Pursuit, the spell explicitly includes the instruction: "You may attach an Equipment with the same controller to it."

Important Clarifications:

• Targeting Requirements: As noted in the RiftJudge FAQ #8437, you must choose both a friendly unit and an equipment you control when casting the spell. Even though the attachment itself is optional (indicated by the word "may"), the equipment must exist on the board to cast the spell.
• "Same Controller" Restriction: As per FAQ #6468, this restriction prevents you from attaching an opponent's equipment to your unit. In a standard 1v1 match, all friendly units and equipment share the same controller, so this restriction is automatically satisfied.

Sequence of Resolution:

1. Cast: You choose a friendly unit and an equipment you control.
2. Move: The chosen unit moves to the destination.
3. Attach (Optional): You may choose to attach the chosen equipment to that unit.
4. Grant Ability: The unit gains the "When I conquer, you may move me to my base" ability until the end of the turn.

Even if you choose not to attach the equipment, the unit still moves and gains the conquest ability.

Note: I could not fully verify the timing of this spell against the retrieved rules.