Canthopexy for Almond Shaped Eyes?

It is possible to perform a canthopexy to reshape the eyes. Almost always, we prefer to simultaneously advance the cheek skin upwards to take tension off the repair. This is called a cheeklift. Our version of the procedure is an ultrashort incision cheeklift, or USIC cheeklift.
A canthopexy or canthoplasty alone can tend to give too severe an upward slant to the eye if the midface tissue is not simultaneously advanced, resulting in too much scleral show in the center (midpupil area) and too severe an uptick laterally.
You should obviously check with a medical doctor to make sure your eye shape has not occurred as a result of any medical condition such as thyroid disease.
Many surgeons are reluctant to operate on patients so young, but if prominent eyeballs detract from the appearance, it may be appropriate to correct their appearance.

You have inadequate bone support along the orbital rim.  This together with your prominent eyes results in inferior scleral show.  TIghtening the lateral canthal tendons (lateral canthoplasty) will no bring the lower eyelid up except at the outer corners.  Rather tightening the lower eyelid will bowstring the lid and pull it further down.  
The answer is a careful personal assessment with consideration for a rim implant and the lengthening of the lower eyelid with hard palate graft.  Carefully study my website below which has a wealth of information about this condition.  In your area you might consider see Marc Cohen an oculoplastic surgeon in New Jersey.

Canthoplasty can be done to improve the contour of the lower eyelid and lateral canthus area.  However, you have be realistic about what can be achieved. You seem to have prominent eyes, which means one have to be careful about what technique to use.  In short, the answer to your question is yes.

Looking at your picture, you have very prominent eyeballs with "white show" of the sclera due to a deficient rim of the lower portion of the eye socket. Any eye surgery in you should be done very carefully as the risk of ectopion (Google picture the term) would be significant. Although a canthopexy would enhance and could be made to elevate the side corners of the eyes, pending a full examination, you may benefit from adding volume to the upper cheek (implant, fat etc) for a more harmonious look. you may need to see a plastic surgeon who specializes in surgery of this area as I suspect the 4 surgeons who turned you down did so for fear of this finding.