Query to join Oracle tables

Honestly, your master table design needs to be normalized. But, in the meantime you could try this query below :

with 
  needed_rows_for_cat1_tab (lvl) as (
  select level from dual 
    connect by level <= (select max(regexp_count(Cat1, ';')) from Your_bad_master_tab) + 1
)
, needed_rows_for_cat2_tab (lvl) as (
  select level from dual 
    connect by level <= (select max(regexp_count(Cat2, ';')) from Your_bad_master_tab) + 1
)
, split_cat1_val_tab as (
      select Entity, Cat1
        , substr(Cat1||';'
          , lag(pos, 1, 0)over(partition by Entity order by lvl) + 1
          , pos - lag(pos, 1, 0)over(partition by Entity order by lvl) - 1
          ) val
        , lvl
        , pos
        , 1 cat
      from (
      select Entity, Cat1, instr(Cat1||';', ';', 1, r1.lvl)pos, r1.lvl
      from Your_bad_master_tab c1
        join needed_rows_for_cat1_tab r1 on r1.lvl <= regexp_count(Cat1, ';') + 1
      )
)
, split_cat2_val_tab as (
      select Entity, Cat2
        , substr(Cat2||';'
          , lag(pos, 1, 0)over(partition by Entity order by lvl) + 1
          , pos - lag(pos, 1, 0)over(partition by Entity order by lvl) - 1
          ) val
        , lvl
        , pos
        , 2 cat
      from (
      select Entity, Cat2, instr(Cat2||';', ';', 1, r2.lvl)pos, r2.lvl
      from Your_bad_master_tab c1
        join needed_rows_for_cat2_tab r2 on r2.lvl <= regexp_count(Cat2, ';') + 1
      )
)
select ENTITY
  , max(decode(cat, 1, CAT1, null)) CAT1
  , listagg(decode(cat, 1, EMAIL, null), ';')within group (order by lvl) EmailCat1 
  , max(decode(cat, 2, CAT1, null)) CAT2
  , listagg(decode(cat, 2, EMAIL, null), ';')within group (order by lvl) EmailCat2 
from (
  select c.*, p.Email
  from split_cat1_val_tab c join Your_person_tab p on (c.val = p.name)
  union all
  select c.*, p.Email
  from split_cat2_val_tab c join Your_person_tab p on (c.val = p.name)
)
group by ENTITY
;

Here are your sample data

--drop table Your_bad_master_tab purge;
create table Your_bad_master_tab (Entity, Cat1, Cat2) as
select 'A', 'Mary;Steve', 'Jacob' from dual union all
select 'B', 'Alex;John', 'Sally;Andrew' from dual
;

--drop table Your_person_tab purge;
create table Your_person_tab (Name, Email, InfoID) as               
select 'Mary',    '[email protected]'     ,'mryD'   from dual union all
select 'Steve',   '[email protected]'    ,'stvR'   from dual union all
select 'Jacob',   '[email protected]'    ,'jacbb'   from dual union all
select 'Sally',   '[email protected]'    ,'sallD'   from dual union all
select 'Alex',    '[email protected]'     ,'alexT'   from dual union all
select 'John',    '[email protected]'     ,'johP'   from dual union all
select 'Andrew',  '[email protected]'   ,'andV'   from dual
;

DB<>fiddle