Its a little more advanced than your example but you could screen scrape your website page and get the value of the textbox that way and then check it against your project textbox.
But you won't find anything as simple as if me.textbox1.text=www.mywebsite.com.textbox1.text then
Is the value entered into the box on the website something that the user does or will it be available when the web page loads? The reason I ask is you could consider making a http request for the page and if the value is there on page load then you should get it back as part of the request. You will then need to go through the data returned to your program in order to find the value and then compare to your application.