I think your function should be as below:

create or replace function search_id (p_key_value in number)
return number
is
found_person number;
begin

select m.ID into found_person from MANAGER m 
where p_key_value=m.ID ;
if sql%rowcount = 1 then
return found_person;
end if ;
exception
when no_data_found then
-- change the field names accordingly
select s.ID into found_person from staff s 
where p_key_value=s.ID ;
   return found_person;
end;

Apex code:

declare
found_person number;
begin
found_person:= search_id(V('P75_RESPONSIBLE'));
apex_util.set_session_state('P75_MANAGER_ID', found_person);
end;

Add P75_RESPONSIBLE to submit items.

You can make the process to run on click of the specific button. You will find the setting in process properties > server-side condition > when button pressed.

Also, you can modify the function to run first update and if update fails then insert. Below is an example:

Create or Replace Function insert_acc_den (p_value in varchar2,
   p_key_value in varchar2)
return boolean
is
begin
update your_table
   set your_column = p_value
where key_column = p_key_value;

if sql%rowcount = 0 then
  -- update fails
  insert into your_table (your_column) values (p_value);
end if;

return true;
exception
when others then
return false;
end;