How to remove and also prevent English Ivy from taking over ?

Pull as much as you can and try Hot boiling water if you cannot use chemicals

Mow it to ground level. This can kill it over time. Pull out any ivy invading your yard and remove the vines. Since English ivy can be started from a cutting, it's important to get rid of the stuff. Moistening your soil can help in pulling the stuff out.

Use a straight white vinegar spray to kill off any of the stuff that creeps into your yard. Another solution is water, salt and liquid soap sprayed on the ivy. Either solution might take a while to do their job, so check back in a few days and repeat the application, avoiding the plants you like.

On hot day try pine soil.

Dogs don't like smell they should stay away from it. Just get the stuff from $1.00 store, its cheap. Did this 45 yrs ago & it worked. But I lived in California at the time & it was 100+ out side.

White vinegar in a spray bottle. Add a good tablespoon or so of table salt, and a spritz of dish washing liquid. spray liberally everyday for a few days on your property. Will not hurt the dog. If he tastes it he may not like it - but it will not hurt him.

Does not answer your 'non-toxic' requirement but safe for pets.

Peel away inch or two of bark, using cling film to wrap around & contain weedkiller granules. This definitely works-used it to clear ivy from the whole of the front of my house-it is not instant clearance but it will die off within the year.

The patch can be put anywhere but preferably as close to the main vine as practical.