Jews of the time understood God to have different roles (and different names), and one of these views of God was as father. For example, in Pirkei Avot, an early mishnaic writing from somewhere between 200 BCE and 200 CE, one rabbi writes:
20 Judah the son of Teima would say: Be bold as a leopard, light as an eagle, fleeting as a deer and mighty as a lion to do the will of your Father in Heaven. He would also say: The brazen--to purgatory; the bashful--to paradise. May it be Your will, L-rd our G-d and G-d of our fathers, that the Holy Temple be rebuilt speedily in our days; and grant us our portion in Your Torah.
(Note that the same passage refers to God the father and our father Avraham. The latter construct appears a lot.)
But this does not cover direct address. Would a Jew of this time address God as "my father"? I found only one use in Tanakh, from Jeremiah chapter 3:
4 Didst thou not just now cry unto Me: 'My father, Thou art the friend of my youth.
Jeremiah seems to be quoting the people as crying out "my father". (Remember that all punctuation is a later addition, but I can't see another way to parse this.)
Evidence from scripture and early mishna seems limited but exists. What about prayer text?
The liturgy was not fixed until much later, but the outline and key concepts were recorded in the talmud, primarily in tractates B'rachot and Yoma. The daily prayer does not contain routine addressing of God as "my father" (or "our father"), but the liturgy for high holy days (and, a later addition, fast days) is rife with it, where in the same passage we say "avinu" (our father) and "malkeinu" (our king). Whether this was yet the case around the time of the destruction of of the temple I do not know.
Conclusion: Textual support is not absent but is not strong. The concept appears in limited form in liturgy. It seems safe to say that referring to God as "my father" was not a routine occurrence in normal speech, but it was also not an unknown idea.
I have addressed here only the use of "father" language, but other petitionary language is found more widely. For example, the torah records incidents of Moshe imploring God and using very polite, deferential language, and Jewish prayer is full of petitionary language. God is not understood to merely be a being on high who cannot be accessed; while God is the transcendant ruler of the universe, he is also immanent and personal.