Not sure what it is, but the non poison removal methods are:

1) Hand pull
or
2) Cover area with 5-7 layers of newspaper, dampen newspaper and cover with 3-4 inches of mulch. This will kill the weeds in 5 to 6 weeks. Use a temporary fence around the area so the dog doesn't dig it up.

In future, keep the area topped up with mulch and hand pull any the weeds when very small.

Hi, It looks like Campanula punctata to me. It sends up long flower shoots, so I could see someone thinking it was turning into a vine. Pulling on it won't help, root just breaks off in the ground. You need to dig. Being under a maple would probably make it hard to dig, but maybe watering real well, and a good hand trowel will do the trick.
Remy

or could be Creeping Bellflower....Campunula rapunculoides. Either way you'll know which if you allow it too flower. Campunula punctata has large ornamental bell flowers that hang down and C. rapunculoides has upright spikes of smaller bellflowers and roots creep to send up more plants. Really a pest when it gets into lawns.
Here the google image results for Creeping Bellflower....do the same for Campunula punctata

Vera

Here is a link that might be useful: Google Images....Creeping Bellflower

I also have a yard FILLED with creeping bellflower. Are there any effective ways to remove it from the lawn, or does one have to dig out the lawn and replace it with new? The flower itself isn't so bad if it is controlled but this yard has been let go for WAY too long!

Thanks!

I have had very good luck using vinegar to control weeds and other invasive plants. I actually eradicated my garden of bamboo by pouring vinegar on it 3 or 4 times last summer. There are still miles of roots/shoots in the ground so I am keeping an eye on it but so far nothing has appeared above ground. I used the vinegar on hot, sunny days when the bamboo was thirsty and it croaked!