It is very common for undergraduates to summer somewhere other than their normal lab, and there are very good reasons for it. In fact, many professors will encourage it.
One of the best reasons is to expand your network. A letter of recommendation from one professor who has supervised your work - great. Two letters of recommendation from two separate professors who have supervised you in research? Better. You also prove that you can work in a lab across different settings and mesh with different people. You may get experience with techniques and processes that you don’t get at your home lab, and you may get access to equipment and research that’s not going on at home. Publications are quite rare for undergrads and I don’t think most of them should be chasing that - I mean, if it happens, that’s great, but it shouldn’t be a concrete must-have goal because you have to enter the lab when the time is just right and know enough to make a significant contribution to the paper. However, you can still get a publication or a presentation out of a summer research experience - I know a few undergrads who have done that.
If an undergrad does 2 years of research during the academic year at their home lab and then spends a summer or two at different labs, they’ll know their home research intimately enough to talk about it while also having a variety of experiences to recount and a wider network of people willing to write letters.
Here’s the thing, though, OP - you can’t wait until the summer. As you see from the REU website, summer student deadlines are usually between January and March of year you want to work somewhere else during the summer. The time to start contacting professors about this would be late October/early November through late March. Also, your email is too vague - you need to be a little more specific, like
Hi, I’m Bobby-Jim McVarnish, a sophomore undergraduate at Westeros University. I’m currently working in Professor Stark’s lab here at Westeros; we’re investigating [something super sciency and awesome here]. I’m really interested in your work on [something else really sciency and also great]; I enjoyed your paper on [briefly discuss a recent paper Professor X has written that you have actually read and liked]. I’d love to work in your lab next/this summer; do you take on summer undergraduates from other universities?"
Worded a bit better than I threw together here, but you get the idea. The professor will know that you need funding, or at least will usually ask you whether or not you do. if they don’t, no need to beat around the bush: directly ask them if they have funding for a summer student in a subsequent email. The worse you’ll get is “no” which is good, because you don’t want to show up and have no way to feed yourself.
Also, it’s not weird at all to do so, but many professors either don’t take summer students or don’t have the funding to take one even if they wanted to. Your best bet is to visit that REU website and apply to formal programs. You can also do an Internet search for things like “summer undergrad research [field]”, “SURF [field]”, “SROP [field]”, etc. (SURF and SROP are initialisms for summer programs.) The upside is that REU/SROP programs usually cover the student’s room costs for the summer and also pay them a decent stipend for food and fun, AND you are in a structured program with other people so you can meet other students in your field. One of my good friends is a person I met at a summer research program, oh jeez, 8 years ago when I was in college. Also if you’re a URM, look up the Leadership Alliance.