Can a missing lower front tooth be bone grafted (built up) for an implant when there is only a few millimeters of space to work in?

Maybe: Depends on the health of the surrounding teeth and bone. This is a question you need to ask the implant surgeon.

Usually yes: But there needs to be enough room to place an implant later on. That should be discussed with your restorative dentist and your surgeon.

Multiple problems: You seem to have 2 problems, bone loss and space loss. You need an Orthodontist to regain space and a Periodontist or Oral Surgeon to then do ridge augmentation/implant placement. Ask your Restorative Dentist to refer you to Orthodontist and to Periodontist or Oral Surgeon for examination, consultation, and collaborative treatment plan.

Room for implant: There needs to be sufficient bone not only to actually place the implant but enough to place a reasonably sized implant & have it sufficiently distant to the adjacent teeth. Trying to "squeeze" an implant in just to replace a missing tooth is not always advisable. Discuss with your own dentist (or a Prosthodontist) who may consult with a Periodontist &/or Orthodontist for treatment alternatives