Well you’d probably want to make it a plugin that registers on certain events.
For one - the call of the page you wish to add the textfield to 'n then to the event responsible for saving/further processing the data after button click.
Maybe you find something on here, it’s the Shopware guide to writing your own plugings: https://developers.shopware.com/developers-guide/example-plugin/
Not sure what sort of text you expect to get from them but maybe you’d want to consider extending the database by your own table “customernumber/id, text” or something like that. In this case your data would not have to depend on the shopware architecture itself too much 'n is also safe from updates, etc.