That's possible, but it's going to take a little (fairly simple) development work. It'll vary a bit by the version you're running, but for most recent 9x versions the ID for that button is 'btnCheckOutNowTop'. You'll want to have your developer remove or comment that out from shoppingcart.aspx, and then go through the codebehind file (shoppingcart.aspx.cs) and remove/comment out references to that button. There are a few other buttons in that area on the page that may or may not display depending on your site config that they may want to deal with as well.
Something to think about - it's usually not considered a great idea to force your customers to scroll down on a page to find the action they're after. Some customers will be confused if what they're after is 'below the fold' and just give up, rather than looking around. It'll vary a lot by your site's design, your customer base, etc, but it might be preferable to put a note near the button you're wanting to remove instead - "Have a promo code? Scroll down!" or something along those lines. Maybe make the change you're after to begin with, then keep an eye on analytics and see if your conversion rates drop.