... show in cell D1, the number in Column C that corresponds to
the first time a 0 will appear in column A following the series of whole
numbers

Try in D1, array-entered (confirm by pressing CTRL+SHIFT+ENTER):
=INDEX(C1:C10,MAX(IF(A1:A10<0,ROW(A1:A10)))+1)

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"Formula help" wrote:
A B C D
0 7 0 9
0 7 6
7 8 5
6 2 0
4 7 6
3 5 5
1 0 7
0 8 9
0 4 8
0 7 2

I have three columns of numbers. Column A will have a series of numbers that
may or may not start with 0s. The 0s will be followed by a series of whole
numbers (non-0 numbers). The whole numbers will then be followed by 0s.
Again, remember that the whole numbers may be preceded by 0s or may not. In
other words, cell A1, A2, etc. may have a 0 or may have a whole number. The
whole numbers will always be followed by 0s.

Columns B and C will have a random series of numbers.

Each time a run a program (import data), the numbers in Column A will
change. The new series of numbers may or may not begin with a 0 but will
always end with a 0 or number of 0s. The numbers in Column B will also
change but the numbers in Column C will not change.

I would like to show in cell D1, the number in Column C that corresponds to
the first time a 0 will appear in column A following the series of whole
numbers. In the example above, 9 is the number in column C that corresponds
to the first time a 0 appears in column A following the series of whole
numbers.

I need the formula for cell D1