Reputable means there will have been observed teaching practice (in a nutshell)
There are, as you can imagine, a million providers these days all offering online training courses. Some of these do offer courses with a teaching practice element and these, while not necessarily
conferring tefli status, will, in today's market (as above, reputable schools are crying out post-Brexit for suitably qualified teachers) probably get you some work. This is F2F obviously. Most bricks and mortar organisations use their existing teachers for any online contracts they pick up.
If the organisation you work for is accredited (in the UK by the British Council) then for every non-tefli teacher employed, they have to produce a written rationale detailing the AQs (alternative qualifications - a PGCE for example) We've just closed our summer centres (and been BC inspected) and had more AQ teachers than ever.
The industry is a bit weird at the moment - highly regulated bricks and mortar schools are having a hell of a job recruiting (mainly in summer obviously when the teenagers come) while the online sector is saturated.
The BC website holds the inspection reports of all the accredited organisations it inspects, and that's a useful starting point to make a list. As is TEFL.com as per pp. There's a Facebook group called online teaching of English (or sth, I'll fetch a link in a mo') that will give you an idea of the good, and the bad, of online.
Also, the BC is now expanding to online teaching and has been creating a "pool" of suitably qualified and experienced teachers for future online work for about a year now. You need to be UK based I think. Note: experienced though. I'm in the pool and have been doing this since 1994.
It's a great job though. I did the CELTA then DELTA back in the 90s and now do teaching, managing, syllabus writing, examining, recruiting, teach in an Italian high school and manage the academic side of a UK summer school. My youngest students have been 2 😳 (only gig I asked to be sacked from back in the day when I'd accept anything) and oldest 79. I prefer teenagers on reflection and only really work with them now apart from the odd uni student.