Can I get a vertical incision abdominoplasty?

The vertical scar is not a substitute for the horizontal scar that goes across the lower abdomen. But it is sometimes added to the horizontal scar in patients who have a large amount of excess skin. This is rare, but does sometimes occur in patients after gastric bypass surgery or in those who lose over 100lbs. It used to be more common when "open" gastric bypasses were common, and patients already had a vertical scar present.

Tummy tuck can absolutely be done with an up and down incision. In fact, this is often done for massive weight loss patients. A more typical horizontal incision is more easily hidden under clothes, but there are some indications for doing a vertical incision.
You should see a plastic surgeon who is certified by the American Board of Plastic Surgery. During the consultation, you should express your concerns about the lower abdominal transverse incision. The surgeon should be able to help you understand the pros and cons of each so that you proceed in the safest way possible. But a tummy tuck through a vertical incision can be done, in the correct circumstances.

In and of its own, a vertical alone is not done because that is not where the majority of patients have issues with as far as skin redundancy and excess goes. Fleur-de-lis abdominoplasty is an option in patients with a lot of loose and excess skin as in patients after massive weight loss. This involves removal of skin in a horizontal and vertical vector.
What I believe you are referring to is just using a vertical incision which in the mind of many is not acceptable when compared to the lower horizontal incision which can be hidden very well, as the vertical cant be.
Hope that answers your question.

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Yes this has been done and is typically combined with a horizontal incision and carries the lovely eupehemistic title the "fleur-de-lis" abdominoplasty mieaning the french lily symbol skin pattern excision.
Most of my patients are terrified of the vertical scar and will do anything to avoid it. Therefore I do not use it commonly. About the only time I use this approach is in a patient with a pre-existing vertical scar which most commonly is from a traditional open gastric bypass procedure.

Removal of abdominal skin using a midline incision is not unheard of at all. It had been proposed over 30 years ago. And it does make more sense from a blood supply/anatomic standpoint. It's a straight shot to repairing the muscles and the skin is easily removed. The only downside is recreating of an esthetically appealing belly button is difficult.