How can you non-surgically remove spider veins?

Sclerotherapy best. Surgery is not an option for spider veins. For leg spider veins the best option is sclerotherapy. Sclerotherapy involves the injection of medications, sclerosants, into the unwanted veins. It works by causing these veins to seal shut. The vein gradually disintegrates. Generally a few treatment sessions are needed. Other options include laser treatment and newer versions of cautery.
A few ways... 1st, find the right "you" to remove them: a board-certified phlebologist. Your doc needs to do a history and exam to see if spiders are an isolated problem or a sign of underlying issues that need to be treated 1st. Spiders are generally injected with sclerosants (many varieties exist, detergent sclerosants are commonly used now). Those too small to be injected can be treated with laser/ipl.
Sclerotherapy. Sclerotherapy, the process of injecting a liquid/foam into the spider veins using a small bore needle is a good way. The commonly used substances are detergents such as polidocanol, sotradecol, morrhuate. Some still use hypertonic saline.
Sclerotherapy. Sclerotherapy (injection of a medicine in the vein) is your best option. It works pretty well. Laser is good also. Treatments are simple and although it takes awhile to heal, the results are pretty good.