Learn Portuguese (Portugal): Há anos que eles procuravam os registos dos antepassados. - They had been looking for the ancestors' records for years.

The structure Há + time + que uses the present tense when an action started in the past and is still happening (which would mean "They have been looking"). By using the imperfect (procuravam), it changes the meaning to "They had been looking" up until some other point in the past.

Could I write this sentence without using que?

Yes, but you would have to change the word order. You can drop the que by moving the time expression to the end: Eles procuravam os registos dos antepassados há anos. Putting Há anos que at the beginning is a special structure used to emphasize how long the action was happening.

Why isn't there a word for "for" after the verb procuravam?

In Portuguese, the verb procurar inherently means "to look for" or "to search for". The "for" is already built into the verb itself, so it is followed directly by the object they are looking for (os registos).

Could we say Faz anos que instead of Há anos que?

While you might hear Faz anos que in Brazilian Portuguese to express elapsed time, European Portuguese almost exclusively uses the verb haver (Há) for these time duration structures.