If you don't hear back from them after a week, you can email Palermo@bytargentina.com directly. This email address reaches their agent, Mariana, who handles that neighborhood. Once we got through to her, everything went smoothly from there.
What we discovered in attempting to request properties through the site is some properties are listed through multiple agents, and, if you request a property & the owner doesn't get back to them, initially we didn't hear back from them. Then we switched to sending them a list of properties that we were interested in that were listed on their website, and Mariana replied with details about which of those properties were regularly rented through them (i.e. which owners were responsive to their queries). Then we chose one, sent the rental request through the website form, and the booking was completed within 1-2 days.
Granted, we haven't stayed there yet, but I've seen others mention ByT Argentina on this forum before, so I have confidence at this point (plan to post a follow up once the rental is complete).
My boyfriend (who doesn't speak Spanish at all) called them via the Whatsapp number listed on their website & they were able to deal with him as an English-only speaker.
Personally, I'd recommend looking through their listings, making the list of the properties you are interested in, attempting to book the top one on the list, and then emailing Palermo@bytargentina.com, if you haven't heard back within a week. In your email, include the property you sent the booking for...and then the rest of the list you made. That way, if the owner of the initial property isn't responding, then Mariana can tell you how regularly they book the remaining properties, and then you can send a 2nd booking request.
Another alternative is AirBnB.com -- much cheaper, much easier, especially since you can pay with a credit card instead of having to bring a lot of USD cash into the country to pay for the rental otherwise. AirBnB.com seems to be the top recommendation on this site.
However, some are not comfortable with AirBnB.com requirement that users upload a copy of either their driver's license or their passport AND use a phone/laptop camera to scan their face to compare to the uploaded ID (you have to move around a little in order for it to accept the scan, so the software can differentiate between a human & a picture of a human). At this point in the rental flow, this isn't even AirBnB anymore, but actually a third party company, Jumio.
Most of the people I know who work in computer security say that everything is already out there (i.e. that these databases already are hacked / shared anyway), so one might as well enjoy some benefits. But everyone decides for themselves.