@Nixwill is correct that you can apply for a Practice Licence as a Payroll Agent or AICB PM.Dip (Associate with Payroll) but in both cases your licence would only enable you to complete payroll services and not bookkeeping.
If you would like to run your own business offering bookkeeping services to clients you need to be a full Member MICB, having completed the Level 2 and 3 core bookkeeping syllabuses in either the current syllabus or the pre-2019 syllabus:
Level 2/II Certificate in Bookkeeping
Level 3/III Certificate in Bookkeeping and Accounts
The Level 4 core bookkeeping qualification (ABA8 and ABA9 exams) has been brought down into the Level 3 syllabus, so you do not need to complete the ABA8 or ABA9 to be eligible to apply for full Membership and a Practice Licence (though, if you're keen you can still complete the exams up until the end of the year!)
@Johnoh and @Brian, as soon as someone has passed their Level 2 core bookkeeping qualifications (new or old) they will be invited to upgrade to Associate Member. As soon as someone passes their Level 3 core bookkeeping qualifications (new or old) they will be invited to upgrade to Member. This means you won't need to request any additional upgrades.
@Johnoh, there are indeed five exams at Level 3 core now compared to four on the old syllabus. This is because the ABA8 Final Accounts for Incorporated Entities exam has been brought down into Level 3 and is now called M7. You'll also notice that instead of taking a centre-based exam at the end of Level 3 (BA7) and again at the end of Level 4 (ABA9) there's now just one centre-based exam (M8) at the end of Level 3.
The overall result being that there are 8 exams to get to MICB compared to 9 on the old syllabus. Whilst you don't need to complete the ABA8 and ABA9 anymore, you may find you are lacking knowledge in Final Accounts for an Incorporated Entity. If you wish to provide this level of service we would advise you to consult your training provider and consider taking either the old ABA8 or the new M7 as per their recommendation/training materials.
ICB will be releasing an M7 textbook shortly!