Possible to move to MBB without an MBA?

100%

You need to network extensively and get your resume as strong as possible.

Good luck!

It is definitely possible to join MBB without an MBA. 
Almost every analyst has no MBA (I mean a serious one, not a fourth year of university). 
Just be aware that if you join an MBB in 1 year, your “seniority” well be likely canceled, and you will have to restart from entry-level. 

Yes of course you can - many people make this move every year. Spend 2-4 years and you will be considered a strong candidate assuming you have done a lot of relevant projects.

I know people who have done that. Their previous experience was helpful for them to enter the recruiting process and get interviews, but it didn’t count towards seniority. They were treated like any entry level new joiner straight out of undergrad

You can absolutely move laterally - many people do this, myself included. You should apply for the entry-level role as you don't have enough experience for the experienced roles. It's great that you have work experience - this will only be a bonus! 

Good luck! 

Yes, it is possible but to maximise your chances:

1. Get referred by personal connections working at MBB; You will increase your chances of an interview significantly.
2. Apply to office in region where you are currently working; your odds are smaller if you are trying to switch firms + locations (e.g. get into London)
3. Reach out for coffee chats to learn more about the local office dynamics

You will only be eligible for an entry-level position normally.