What should I do if I'm trying to go into academic from a mostly non-academic background? I have finished an undergraduate programme in Biology, and went on to do a Masters in Global Health. 

The problem is that I don't really have any articles of thesis to show. My undergraduate programme didn't have the option to do a thesis (I assumed all undergrad programmes did, so I didn't look into it before joining). I did a thesis for the Masters, but it wasn't that great. My supervisor at the time took on too many students and couldn't give us proper directions. I guess he assumed we knew what we were doing, but I hadn't had previous experience so I had no clue (he was also banned from being a supervisor for my course afterwards as a lot of students under him were failing due to lack of guidance). I actually finished it, passed and managed to graduate, but it is not good and I wouldn't submit it to a new study. 

Needless it is to say, I also don't have any letters of recommendations. I'm quite introverted and, although there were a few tutors and lecturers who knew me, none did any major works with me and I didn't ask for letters of recommendation.

I'm currently working at a non-academic field, to do with Health Policies, but I want to go back to an academic field. I'm at a loss of what to do though. 

I was wondering if it would be possible at all to do a PhD or research position (mostly because there are more paid positions or grants for these), or if it would be best for me to do another masters (only reason why I hesitate is that they are very costly). Or maybe I should approach some old teachers and ask for recommendation letters? I could ask some from my current manager at work, but then he would take it as me wanting to quit my job, not a good move until I do have something lined up. 

Anybody has advice? I'm based in Europe, btw. 

TLDR: have an undergrad and masters, but no (quality) thesis, and no letters of recommendation. Working a job, want to go to academia. Which routes could I take?
Generally, I do not believe that it will be problem that you do not have an extensive Master thesis. Most universities are likely to pay more attention to your grades and other documents such as letters of recommendation, motivation letters, etc.
So, the letters of recommendation could be a problem. I would advise reaching out to your old professors and tutors. Focus on those from whom you received good grades, briefly explain your reasons for reaching out now (in 1 or 2 sentences), and mention that you got a good grade in their course. At least in my experience, most professors/tutors will be willing to help. If it puts you in an uneasy position in your current job, then you should not ask your manager a letter of recommendation. This can easily backfire. However, it does not have to be your manager writing the letter. If possible try a more senior colleague whom you can trust.
If you apply for a Ph.D. position that is associated with a specific project, then make sure that you do your homework on the topic and show it in any application letter or motivation letter. The same holds for any position that associated a specific advisor from the beginning. In this make sure you know the advisor's work and research agenda.
If you feel not ready for a Ph.D. you could also consider programs that include coursework in the first year. Overall, do not overthink it, and do not forget that a Ph.D. program is there to teach you how to do research. In your application focus on conveying that you are able and willing to learn how to do that. Your professional experience could be a plus and more important than the perceived quality of your Master Thesis.