Two ways you can approach that:

• each observatory has their own archive interface (you can just search for "jwst archive", "hubble archive", "eso archive" etc.)
• some observatories implement so-called Virtual Observatory protocols, which in principal allows to query all of them using the same code or the same tools (there are some tools like Aladin Sky which support those)

Depending on your goal either of those might be more useful. First option provides much more in-depth tools and is useful if you're looking for specific type of data - eg. you want data from particular instrument. Second option is useful if you're interested in data about some specific target, and you want to get all available observations of such target.

There is also one more thing to consider -> archives provide two types of data:

• raw data
• science data

Raw data means essentially a direct read-out of the instrument which requires complex multi-stage processing and combining with other data (aka: calibrations) before you get something useful. This is useful if you are developing some special processing steps, or if you know something particular about the target and you can configure parameters for the processing better than the default ones.

Second option is more useful if your intention is just to search and analyse science data.