First of all, let's be clear, unless you are made of money, a quick wedding is not going to look like a wedding that took a year to plan. That being said, it can still be wonderful and sweet and amazing. We'll help you plan a terrific event on a last-minute schedule.
You're going to need someone to make this thing official. If you are religious, contact your church's leader and see who could be available to you on such short notice. Some religions don't permit weddings outside of the church or temple, so you should also look into a private officiant. If you want, you can also have a friend or family member licensed online within a day in most states.
Especially if you are trying to plan your elopement in the summer, most of the traditional wedding venues will be booked. Check into local parks, favorite restaurants, and even unconventional locations like a miniature golf course or carnival. A restaurant or cafe will probably be the simplest choice, so that you can do in-house catering (which will save you from having to search down another vendor).
There may not be time to order a traditional cake, but most bakeries will be able to accommodate layer cakes or other desserts for a quick wedding. And keep in mind, that etiquette dictates that you should only offer some type of refreshment to your guests. But that can be tea and a slice of pie. So, don't stress yourselves out over large expenses for food and drink.
More than likely, you're going to have to DIY these to some extent. Talk to your florist about what they have the time and ability to accomplish. Fill in the rest with loose flowers on your own. Or skip the flowers altogether in favor of the venue's everyday decor.
You can order personalized invitations quickly, but that's still not going to get you the responses you need in time. Instead, opt to send an online invitation or link to a wedding website. If you want to send out something hard copy, wait to send wedding announcements when you return from your honeymoon. You can send these both to your guests and people that didn't get an invite.