I have a process which writes a file to be bcp'd in. When I run the
bcp command I am using the character format (-c option), and I am
wondering about the behaviour when I encounter numeric values
represented in scientific notation.

I have noticed that for a column that is defined as "numeric(21,4)" I
can manually insert a value that is represented in scientific
notation, however in the bcp, I get an overflow error.

Is there any way that I can make the bcp command behave the same way
an insert statement would, when encountering scientific notation?

which version of ASE and BCP you are using ?
and how does your bcp file looks like ?

make sure your whole string should not be bigger than 21.
for example:

value --> 123456789123456789123.1234 --> not working
value --> 12345678912345678.1234 --> working

have you tried to increase the length of your numeric field?