Excel has a function called CheckSpelling(word) that returns true if the word is spelled correctly, or false otherwise.  Access doesn't have this feature, but thanks to automation, Access can use it like any other feature.

See picture below.  First you need to add the Excel Object Library to the list of "references" in the Access VBA editor.  Then you create a custom function MySpellCheck, as shown, that calls the built-in Excel function CheckSpelling().  Basically, Access "declares" a new instance of the Excel application, and grabs a function from Excel as if it were its own.  

Then you can use your custom function anywhere you want, in queries or other VBA procedures.

One caveat is that if there are multiple words, the CheckSpelling() function returns false if ANY one of the words is misspelled, and doesn't tell you which one.  If you need it to work for individual words, you need to split up any multi-word string into individual words, with the VBA function Split().  For instance:

Dim arr As Variant
arr = Split("This is a test.", " ")

"arr" is an array variable, and the above yields: arr(0) = "This", arr(1) = "is", arr(2) = "a", arr(3) = "test."

And finally, to achieve your end goal, you need to run a query to put misspelled words into a table, with DoCmd.RunSql or CurrentDB.Execute.

This is not a terribly difficult problem, mainly because of how easy it is for Access to grab the spellcheck function from Excel.  But you do need skills and knowledge from several areas: automation, VBA, Split(), arrays, queries, DoCmd.RunSQL, CurrentDB.Execute, etc.  Hopefully the other posters could steer you to the right courses if you need them.

Thank you Kevin!! This is great stuff!  I do have long text fields, so not sure if splitting them into many, many individual word fields makes sense, but would love any other thoughts anyone has on a way to make this work in a workable way.
Thanks again!

You don't need to split them into fields; your table structures and data aren't affected at all.  You just split them, and store them into an array variable ("arr" in my sample code above), and use VBA to check the words in that variable for misspelled words.  Sample code:

Dim arr As Variant, n As Long, i As Long
arr = Split(MyLongText, " ")
n = UBound(arr)    ' n = number of words minus 1
For i = 0 To n
     If MySpellCheck(arr(i)) = False Then DoSomething
Next i

To log a misspelled word, a simple append query to another table would do it.  That table would need to be a child table in a one-to-many relationship, because each of your long text field could have many misspelled words.  This site has several videos on append queries and one-to-many relationships.

Thank you and thank you!  This seems doable.  Really nice clear instructional help :-)
Best,
Gary