Realtime
If you visit the realtime tracking page, you can click on your institution name to access the data (.csv) and the link in the status column to access the map (.html). If there is more than one track in the file, you can search by Deployment ID, which will be emailed to you.
Historical
At least once per year drifter data are reviewed and cleaned. This reviewed data (known as Quality Assured/Quality Controlled data, QA/QC) is available through ERDDAP. ERDDAP (previously an acronym, now just the name) is a data server that gives you a simple, consistent way to download subsets of scientific datasets in common file formats and make graphs and maps. To access the Drifter ERDDAP tool you can normally visit : http://comet.nefsc.noaa.gov/erddap/tabledap/drifters.html however, in mid-2022 this site went down for upgrades and, as of Oct 2022, it is not yet back up.
In the meantime, to access drifter data, there is a simple ascii file at https://studentsdrifters.org/drift.dat but this is a huge file with 1500 tracks that is difficult to manage.