> the problem is that I'm trying to put a "hat" over a consonant
I suspect you'll find that, the more precise and specific your question, the more greater the likelihood of getting a speedy and helpful answer.
> Looks like the software only allows the circumflex over vowels.
> Option-i does work on my computer for vowels. (Built-in
> keyboard).
Nonsense. Nothing to do with your keyboard. Btw, you still haven't told us what character you're trying to insert.
> I guess I will have to use the Equation Editor.
That's an option, but not the only one. Just for the sake of argument, let's assume it's p-hat (not the party hat, but the symbol for sample proportion).
(1) Specifically in Word, you can use fields (overstrike). Go to Insert > Field… > Equations and Formulas > Eq; then type
EQ \O (p,^)
and confirm. (Note that "EQ" will be inserted by Word automatically, you don't need to type it again.)
Once you get the hang of it, you won't need to insert it using menu & submenu commands, but by typing and applying styles.
(2) The right way is Unicode. Currently, Unicode defines >100k codepoints, so, for obvious reasons, it's neither necessary nor practical for most keyboard layouts to be able to access anything but a small subset. You can enter Unicode characters using Character Viewer (aka Special Characters), or, more efficiently, using the Unicode Hex Input keyboard layout.
If the character is already defined as such in Unicode, eg, for all (upside down A), then hold down Option and type its Unicode hex code, ie 2200 -- ∀. However, p-hat isn't defined as such, so it must be entered as a combining character sequence. The sequence is ((Latin small letter p) + (combining circumflex accent)), or (p + 0302) -- p̂.
Unfortunately, all is not sweetness and light. Although this is how it should be done, the display may disappoint. Questions 12b and 12c in the respective Unicode FAQ explain what happens and why.