The short answer is yes, you can have orthodontic treatment with an implant already in place. The useful answer is about sequence, because there is one property of implants that governs everything else.
An implant cannot be moved
A natural tooth is suspended in its socket by the periodontal ligament, a thin layer of fibres that allows the tooth to shift slowly when steady pressure is applied. That is the entire mechanism orthodontics relies on.
An implant has no ligament. It is fused directly to the bone, a state called osseointegration, and that is precisely what makes it strong. Apply orthodontic force to it and nothing happens. It is, for planning purposes, a fixed post in the jaw.
What that means in practice
Your orthodontist has to design the finished position of every other tooth around wherever the implant already is. If the implant sits close to where it should be, that is a minor constraint. If it was placed years ago in a position that no longer suits the corrected arch, the options narrow: accept a compromise in the final alignment, or remove and replace the implant, which is a significant undertaking.
There is one advantage. Because it does not move, an implant makes an excellent anchor. Orthodontists sometimes place small temporary anchorage devices specifically to get the immovable point an implant already provides.
The right order, when you have the choice
Straighten first, implant second. Almost always. Moving the natural teeth into their final positions creates the correct space, then the implant is placed into a gap of the right width with the right amount of bone on either side. Doing it the other way round means building the alignment around a fixed obstacle.
If you are currently missing a tooth and considering both, raise the orthodontic question before anyone places an implant. It is a much easier conversation before than after.
Holding the space in the meantime
The gap still needs managing during treatment. Aligners can carry a tooth-coloured pontic, effectively a false tooth built into the aligner, so there is no visible gap while you wait. With fixed braces, a temporary tooth can be bonded to the wire. Either way, the space is kept open at the correct width rather than being allowed to close.
A note on crowns and bridges
Crowned natural teeth move normally, since the root and ligament are intact, although brackets and attachments bond less reliably to porcelain than to enamel and may need the surface preparing. A conventional bridge, on the other hand, splints the teeth it joins and usually has to be sectioned or replaced before they can be moved independently.