I know Amazon have been mucking about with their music streaming service but I've not tried to buy a download from them for ages because they only do mp3 and not any lossless format like FLAC. However, there are places to buy downloads. Here are a few I use, but I always buy FLAC so you'd have to check for mp3 availability (although there are free apps to convert).
Qobuz is the world leader in 24-bit Hi-Res downloads, offering more than 100 million tracks for streaming in unequalled sound quality (FLAC 16 Bits / 44.1kHz)
www.qobuz.com
Basically a streaming service but has a download store.
7digital provides a catalogue of over 30 million high-quality music tracks and a wealth of uniquely curated recommendations. Browse, buy, listen and download!
uk.7digital.com
Bleep - We specialise in independent and innovative music regardless of genre or format. Working with labels and artists around the globe, we bring you a hand-picked selection of music for your ears and your feet.
bleep.com
Also, great place to discover new and up-coming music. Stream whole albums for free before you buy. Search by genre. Purchase gives you both unlimited streaming and downloads in multiple formats, including mp3: You'll not find major artists there though. I know this sounds a bit like an ad, by I have no connection to them other than an enthusiastic customer.
Discover amazing music and directly support the artists who make it.
bandcamp.com