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;

The issue seems in your if condition. You are comparing as:

if v_state = v_state

which is wrong.

I think, it should be:

if v_state = v_stored_state

Also, after end if; return the v_page. Following is the complete code:

declare
v_state varchar2(50);
v_stored_state varchar2(50);
v_page number;

begin
v_state:= V('STATE_OF_FORM');
select fp.STATE, fp.PAGE into v_stored_state, v_page from PROCESSSTEPS_FORM fp where fp.AUDIT_USER=:APP_USER;

IF nvl(v_state, 'x') = nvl(v_stored_state, 'y')
THEN
SELECT APEX_PAGE.GET_URL (
p_page => v_page)
INTO v_page;

FROM DUAL;
end if;
return v_page;
end;