Hi Sarah,
Sadly no bus makes it from there. Ste-Mere-Eglise is serviced by the Maneo bus network. There is no train station in Ste-Mere-Eglise either, but it is just 3.5 km (2.17 miles) from the Chef-du-Pont train station... though this train station is very infrequently serviced.
If you take a look at the Maneo service map, you can see that Ste-Mere-Eglise is on the number 1 route (in red) just south of Valognes.
Perhaps you can go from Bayeux to Carentan via train (I confirmed there are regular trips that are 25 minutes in duration), from where you can take the bus the rest of the way (another 23 minutes). The schedule for bus line 1 through Ste-Mere-Eglise is not very frequent at all, so make sure you are well aware of the bus times in advance.