Can you say searching happiness or do you have to say searching for happiness?
The 2nd one.
If the sentence is 'I am searching', the sentence is ok as it is.
As soon as you add the object of the verb 'to search' a preposition - 'for' is required, as in 'I am searching for..... my husband/ a better life / the dog '
It would be 'Happiness is something that all people are looking for.' If you would like to include the it, you could instead use a semi-colon to link the 'it' into a new sentence. For example:
'Happiness is something that all people are looking for; it is a psychological need' or something to that effect.