I think that you can find the answer to your question here: trigger datalist options on click event of text box (your question is possible duplicate).
My made-up combobox is missing its arrow
Question
I have a made-up combobox... more like a textbox with suggestions... in chrome, it has an arrow, making it like a real combobox... but this arrow is missing in mozilla. is there a way for this arrow to also appear in mozilla? I really need it as a combobox and not a drop-down list.
I have no special css for it but here is the code:
<input type="text" list="pilian" id="gi_classification" name="gi_classification">
<datalist id="pilian">
<option>For Sale</option>
<option>For Lease</option>
<option>For Sale & For Lease</option>
</datalist>
Solution
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