It is not possible to insert a checkbox directly into a <select>
or multi-select HTML element.
For alternative options, you can refer to this related question where various solutions are discussed.
The most practical solution that may suit your needs can be found here:
If you want to change the default checkmark to an X, you can follow this example:
Additional resources:
How to use Checkbox inside Select Option
After selecting check box Instead of Tick symbol need X in html
UPDATE:
If you require both the HTML markup and javascript to achieve your desired functionality, you need to follow these steps:
Refer to this answer for an example of creating a custom <select>
control.
Learn how to replace the checkbox from step 1 with an icon/image of your choice by checking out this answer.
The javascript to remove the X'd <option>
is quite straightforward:
<form>
submission.
If you need assistance developing the solution, consider utilizing services from freelance websites - personal recommendation based on positive experiences.