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This is definitely possible. First create a file field to attach to your order in the control panel. Then create a file input type with a name attribute that uses the field you created name="fields[fieldName]". This would then get sent in the post request to updateCart.

<form method="POST" enctype="multipart/form-data">
  <input type="hidden" name="action" value="commerce/cart/updateCart">
  {{ getCsrfInput() }}
  <input type="file" name="fields[fieldName][]">
  <button type="submit">Upload file</button>
</form>