How to connect a button to html buttons

You should use Toolbox Plugin | Bubble plugin to send data from your HTML button to Bubble. Specifically, you have to use Javascript to Bubble element. See the demo below that gets the input’s value inside a button in HTML and send it back to an alert:
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Simply, this element produces a function available to trigger an event in Bubble:

After setting these, you have a function that you can call in any HTML element. See how I call the function bubble_fn_try with some parameter (the value of the input box):
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The first checkbox Trigger event mean they would trigger a workflow in Bubble:
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The second checkbox Publish value is to receive parameters. This is how I showed the value in an alert for example:
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Of course, you should also set the Value type to text (or whatever type you need) so it will not give errors when you are getting the value. Otherwise, you will get this error:

For the full example, check this editor: Tests for Forum 10 | Bubble Editor