How would I make an audio merge site using no-code tools?

The easiest way to do something like this would be to use another 3rd party API tool to combine the files. So it depends on your budget and knowledge of APIs. Bubble can do the external API calls and then update after processing is complete.

One example is Cloudinary, audio is treated the same as video transformations so don't get confused by that: https://cloudinary.com/documentation/video_manipulation_and_delivery#trimming_videos

There's more APIs out there too, just have to find one that meets your usecase.

Warning: most APIs that you'll have won't let you combine one audio file or codec with totally different one, so you'll have that step in there too if someone gives you a really high quality MP4 and someone else gives you a WAV file you'll have to have an extra step to identify then transform it to your common file type, then go through and do your audio edits/combining. That increases the steps a bit and user error handling but also increases your price (see below) of providing the service.

Warning 2: audio (and video) is priced per minute, so it can get expensive quick if you get a lot of audio coming together from many users. Example: if for some reason you used Google to grab transcription, you'd be charged per min, so a 5 min clip would cost you 5*$0.024USD/min = $0.12 or a full 60 min clip combining it would be $1.44; use case I would say is possible -- let's say your users use a service to record meetings they have for work; then you provide a transcript of it for notes and records -- if they had 10 meetings a month for 1 hour each, you'd be out $14.40USD/month to have them as a user at most -- just make sure that you don't under charge for your processing fees.

Source: Google Cloud Text To Speech Pricing