First of all determine if you have field bindweed or hedge bindweed. Google each for images. Hedge bindweed is a little easier to get rid of.

https://www.minnesotawildflowers.info/flower/hedge-bindweed

Bindweed is one of the most difficult weeds to eradicate. It can put down roots as deep as 10 feet. Its also difficult to control due to its long-lived seeds and extensive root system that can re-sprout after most management actions. The key to managing field bindweed is depleting root reserves and reducing its re-sprouting ability, which takes time and persistence. Mechanical control methods need to be repeated every couple of weeks during the growing season for several years to effectively remove field bindweed shoots and roots. Mowing is not considered effective as field bindweed is a low-growing plant. Pulling, burning, and mechanical cultivation may be used but should be repeated often or integrated with another management approach, such as herbicides, in order to reduce field bindweed.

Manually: To get rid of bindweed, use a pitchfork to gently turn over the soil underneath the plants and pull them out by hand. Try to pull the whole root out, which can grow as deep as 10 feet, to prevent new shoots from growing. You will need to repeat this each time new sprouts pop up. This is the safest method for your arborvitaes but also the most back-breaking.

Almost any method you use (other than continually cutting it back to ground level every time it sprouts up) is going to harm the roots of your arborvitae. That includes boiling water. Think about it, if boiling water will kill a bindweed root (which requires applying the hot water 2-3 feet in a circle around the plant), won't it also kill any other roots with which it comes into contact.

I don't like to recommend chemical control unless absolutely necessary. It is an effective method to use a systemic herbicide that moves into the root system to kill roots and root buds. Herbicides can be applied in spring or summer, and it is usually recommended to spray when bindweed is in the bud to full bloom stage. Herbicides labeled for field bindweed control include 2,4-D, aminocyclopyrachlor + chlorsulfuron (Perspective™), dicamba (Banvel®), glyphosate (Roundup®), imazapic (Plateau®), metsulfuron (Escort®), picloram (Tordon®), and quinclorac (Paramount®). Refer to specific herbicide labels for complete information and recommendations. Any one of these will likely require repeat application. You will need to thoroughly protect your arborvitae from the spray or drift. Do not spray on a windy day. As always read and follow the label instructions; keep pets away; and wear protective clothing and gloves. A note on glyphosate: this chemical has been linked to cancer claims, so be aware. In spite of your best efforts, it is going to be difficult to avoid at least some damage to your arborvitaes.

Here is some additional helpful information:

https://blogs.cornell.edu/weedid/859-2/