Henderman said: If you're willing to spend a few bucks hire someone with a bobcat to come by and dig it all out in one shot*. It'll cost you some bucks, but it'll be done quickly and the bobcat can dig down and get a lot of the big roots. Be sure to tell them about the poison ivy before they start.
Having driven 5 different makes when I landscaped , it can dig it out but then you will be replacing a LARGE portion of your lawn, and you will have to have replacement dirt hauled in and leveled out, with scrap dirt hauled away.
It is NOT a one shot deal, to get roots you must go down well over a foot; skid-steer buckets do not dig like a shovel .
Also, most areas are fussy about the scrap dirt with poison ivy, in Minnesota it is specially regulated.
Skid-steers are wonderful tools for many tasks but are often far more destructive to surrounding areas that people realize.
Addendum:
Looking at the photos, IF, if a skid steer came in from the street side, and just leveled he bucket on the sidewalk, it could just keep the bucket down and keep going straight till it reached the far side of the garden.
There would be some boulevard grass needing fixing, and make sure the side walk is not ruined but in your case it could be a relatively simple job. (That does not mean cheap.)
Removal and replacing dirt would be one half the cost, and up here dirt is cheap.
IF the poison Ivy were already killed , the person doing the job would be much happier.
In one weed whacking job we had once, the person who hired us wanted me to weed whack an area full of Poison Ivy, I said NO, I will not do that area, never got pushed any farther.