Append text to column data in PostgreSQL if columns initial value is null

It's pretty unclear what you are trying to achieve, but:

If the column's value is null you can't "append" a value to it because any expression involving null yields null ( null||'foo' is null). In this case you just replace the null value with the new value:

update the_table
  set the_column = 'new value'
where the_column is null;

If with "initial value is null" you mean if the "current value is an empty string", then you would do something like this:

update the_table
  set the_column = the_column || 'this will be appended'
where the_column = '';

Which is identical to:

update the_table
  set the_column = 'this will be appended'
where the_column = '';

null and '' are different things

Another option is to use the concat() function which will implicitly treat null values as empty strings:

update the_table
  set the_column = concat(the_column, 'this will be appended')
where ...