WMA files are usually inherited rather than chosen: a CD collection ripped with Windows Media Player in the 2000s, when WMA was the default and nobody thought to change it. The files were fine on that PC and are awkward everywhere since — no Mac, no phone, no browser and no modern car stereo plays WMA without help. Converting the library to MP3 is the standard way to make a decade of ripping usable again, and it is the kind of bulk job you want running on your own machine rather than uploaded a folder at a time.
Some WMA files are locked and no converter can open them
This is the one thing worth knowing before you start. Music bought from the subscription stores of that era — Napster, Yahoo Music, Zune Pass, MSN Music and the rest — was wrapped in Windows Media DRM, which encrypts the audio and ties playback to a licence file on the machine that bought it. The extension still reads .wma and the file still shows a title and artist.
Those files cannot be decoded here, or by any other converter, and the conversion will simply fail on them. That is not a limitation of running in the browser; the audio is encrypted and the key is not in the file. Tracks you ripped yourself from your own CDs are never affected, and neither is anything bought after those stores closed. If a file fails while its neighbours succeed, DRM is the likely reason.
What WMA actually is, and which flavour you have
A .wma file is an ASF container, Microsoft's multimedia wrapper, holding one of four different codecs. Standard WMA is the lossy one nearly every ripped CD used. WMA Pro added multichannel and higher sample rates. WMA Lossless stores audio bit-for-bit like FLAC. WMA Voice is a low-bitrate speech codec used for dictation.
You do not need to identify yours — all four decode here — but the flavour decides what the conversion costs you. A standard WMA rip is already lossy, so the MP3 is a second pass and slightly worse than the source. A WMA Lossless file is a perfect copy of the CD, and converting it to MP3 discards audio you currently have, so keep the originals if they are your only copy.