Hi @lukasbeule,
I can come up with four options for you, depending on your project size and additional security concerns.
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You could try an Amazon Web Services EC2 instance. This would let you access your work from a browser, but would mean hosting your data remotely in the cloud. This costs money; I'm not sure about the status of project grants. I would double check the ethical implications of this as well; your ethics board may not allow you to do this.
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Look into regional or national compute infrastructure. I've worked in the US and Europe, and in both places, we've been able to buy time on a departmental/institutional/state/national super computing resource. This may be more power than you need. Typically, these computers have better GDPR assurances if you're working with sensitive data (for example, our Swedish resource has a dedicated system for genetic data.) This would require something like a PUTTY interface to access.
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It looks like qiime2 has moved into galaxy (q2Galaxy - Galaxy Community Hub). I haven't been involved, so I'm personally sure about the status; some other moderator may be able to tell you more. I also don't know what this means about your data, if there is a local concern
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Qiita offers some analysis capabilities, depending on what you're doing. Your data would be hosted at UCSD and the team at UCSD would be able to see your metadata. There may also be a time limit during which your data can be kept private, although its been a while since I worked directly with Qiita. You are also limited to the annotation methods/database/visualization in the database.
You are far more patient than I am.
Best,
Justine