It's important to understand the difference between audio streaming and audio on demand.
Audio Streaming: Is audio that is live or pre-recorded programing that is being played on a set schedule. For example the output from an online radio station.
Audio Streaming platforms/servers include Wavestreaming which feeds Tune-In Radio and other apps. More recent platforms like Facebook live or Youtube Live can also be considered as streaming. Some radio stations build their own streaming server but most simply purchase a 3rd party platform to avoid the headache of hacks/viruses or maintaining the servers.
Audio On Demand: Is audio available online or on a device to listen to at any time. For example an audio file that is put on a website or audio uploaded to a podcast server.
Most websites can host audio, but programers try to avoid that so as not to make the website so 'heavy' and difficult to load. Plus with increased social media sharing, it is best to use a third party app such as Soundcloud or a podcast server.
Podcast servers are Buzzsprout, Libsync, Audioboom and so on. The advantage of using a podcast server is that, once you have uploaded the audio, it is sent to iTunes, Google Play and other podcasting apps.
Audio streaming servers like Wavestreaming have a built in a mobile application tool that allows you to create a personalized app that runs of their server. You have to pay the programing fees. Last time I checked, I think it was $150 a year. (iTunes require all programers or app creators to pay a yearly fee for their apps)
Podcast Servers start at $120 a year and go up to $3,600. The smaller servers charge you based on a flat fee, some larger servers charge the flat fee plus a charge based on bandwidth used or space used.
Mobile Apps can cost anywhere from $500 to $5,000. Here is a really good article that explains the process. Pay special attention to Waterfall and Agile. Most people think they know wha they want (waterfall) but in reality just have a foundation for what they want and end up adding more and more things to the app (Agile) which end up costing them. Source: https://www.applicoinc.com/blog/much-cost-build-app/
The cheaper app developers can easily be found on Craiglist, where they offer a template app for a business that cost $100. The price usually goes up, if you are streaming audio. Then there are companies in places like Los Angeles that will work with you. I don't know what the cost of that is, since they never wanted to give me a ballpark figure. You can also work with someone that specifically works with radio stations such as Lee Harris from http://harrismedia.com/ who understands radio and can possibly customizes something.
It's important to keep in mind that launching an app is only the first step. Maintaining and updating meta data is key to gaining new listeners. Of course if you already have a large fan base it is a matter of seeing what is the most affordable way of distributing your audio to as many platforms as possible.