You don't give your USDA hardiness zone so it might prove to be a challenge for folks to give you ideas without that information.

I don't have clay soil but creeping myrtle/Vinca minor seems to thrive where I am in sun, shade and everything in between. Google search says it's hardy in Z4-9. It's been growing here since about 1965 or so and is evergreen.

I Googled hoping to get soil preferences but didn't find anything that helped.

Pachysandra is another great evergreen groundcover although it prefers soil that isn't compacted the way clay soil is. It also prefers shade but I've seen it growing in sun as well..

Here is a link that might be useful: Pachysandra

You really have two separate issues, the grub problem and what to grow as a groundcover. Is this in place of the lawn for the whole yard? If so, grass is probably easiest as far as maintenance if you deal with the grub issue. Milky spore disease or nematodes are two nontoxic ways to deal with grubs, and if you are in the warmer end of New England Milky spore will involve one application that takes a couple of years to build up to good effectiveness. I imagine that the grubs will cause problems for other plants than grass, though I am not sure that would be the case for all plants.

I always work better from a photo or at least a description of the area: size, growing zone, how much work you are willing to do, aesthetic preferences. Do you want the look of all the same or do you want a variety so that it looks more like a garden? Are you willing to put in some type of barrier and do the work needed to keep the ground cover from migrating from your yard to the neighbors' if you share boundaries? (In the case of aggressive running groundcovers, good barriers make for more friendly neighbor relations.) Are you open to shrubby groundcovers? What kind of ordinances does your city have about lawns? What is the drainage like (are you on a slope, does water pond in the area you are considering, have you done a percolation test?) Does it have to be evergreen or is a plant that dies to the ground in winter OK, especially if you renew the mulch after freeze to keep it tidy looking?

Some general comments and suggestions:
Getting rid of the grass and controlling weeds for the time until your groundcover gets established will be a lot of work. For many groundcovers, mulch will be your friend.
Improving the soil with a combination of coarse and finer organic matter may open up more plant choices, depending on how much clay is in your soil and what the slope is like.
My first thought was also periwinkle/myrtle/Vinca.
The house I grew up in had Pachysandra growing in clay in sun until shortly after noon, but had bright shade from the house in the hottest part of the day, so I am not sure how this will do. Evergreen
Bugleweed/Ajuga reptans will grow in full sun, and isn't fussy about soil. Evergreen, flowers, colored foliage.
Bishop's weed/goutweed/Aegopodium podagraria has a variegated form that is a good groundcover that is tolerant of many growing conditions. My inlaws have it growing on a dry sunny slope, but I have also seen it in half day sun on clay and in full shade.
If drainage is good, some sedums, thymes, creeping cottoneaster (woody), creeping Juniper (woody, evergreen), or one of the lower growing Nepeta/catmints might work.
Cranberry doesn't need a bog to grow, just averge moisture, so if your soil is acid it might work with mulch. Evergreen
Mazus reptans might work.
Bearberry/Arctostaphylos uva-ursi 'Massachusetts' (woody, evergreen)
Cutleaf Stephanandra/Stephanandra incisa ‘Crispa’ woody, fall color
In a smaller area, perennial candytuft (Iberis sempervirens) may work. Evergreen.
If a taller groundcover is OK, and you don't mind a area that is bare of plants in winter, it's hard to beat old-fashioned orange daylilies. The photo is of a daylily bed just starting to sprout in the spring. The mulch is renewed after die-back in the winter and helps control weeds and erosion and keeps things looking tidy when there's no snow.

Once again, sounds like an answer to be preserved for the future. You do give great answers, nhbabs.

I use daylilies and sedum in those dry clay areas and over the years, after applications of mulched leaves in the fall, I now have a bee garden full of monarda, lilies and coneflowers.

Jane

I second (third? fourth?) the idea of dropping daylily bulbs to establish the bed for further plantings. They're VERY easy to move (just dig up the bulbs) and at least the ones in my yard tolerate being moved very well.

I'm in the process right now of converting grass space into a garden bed. The first thing I did was laid down cardboard, to shade the grass and encourage worms. Then I laid down as many daylily bulbs as will fit in the space. I'll cover them with half finished compost, finished compost, and some year-old soil the next time I'm home in order to finish the plantings. Can't wait to see them come up next year.

And if it's a failure: the hundreds of other daylily bulbs that grow like weeds throughout my beds will make nice replacements....

I've used Arctostaphylus uva ursi and I like it a lot. It spreads very well and never needs attention and is very pretty. It offers berries for wildlife, is a native and doesn't become a nuisance.

Personally, I wouldn't use Vinca in full sun. I do use it but it is in shade and is bordered by the street on one side and the driveway on another and a fence on a third. I tried it in full sun once and after one season it had spread so much that it scared me and I ripped it out.