According to Jewish sources, do animals have a connection with God?
by connection I mean some kind of awareness of His existence or perhaps some sort of communion.
Psalms (104:21) states:
הַכְּפִירִים, שֹׁאֲגִים לַטָּרֶף; וּלְבַקֵּשׁ מֵאֵל, אָכְלָם
The young lions roar after their prey, and seek their food from God.
At face value this implies that on some level the animals commune with God and express their needs to God.
This seems to be the understanding of the Metsudat David (there) who explains simply:
שואגים לבקש מאל אכלם
They roar to request their food from God.
Similarly, Psalms (145:15) states:
טו עֵינֵי-כֹל, אֵלֶיךָ יְשַׂבֵּרוּ; וְאַתָּה נוֹתֵן-לָהֶם אֶת-אָכְלָם בְּעִתּוֹ.
The eyes of all wait for Thee, and Thou givest them their food in due season.
This statement that all look to God for sustenance, could be interpreted to include animals as well. Particularly, in light of the next verse that speaks of God satisfying כל חי; every living thing.
Indeed, this seems to be the understanding of R. Elazar Rokeah who writes in his commentary to the Siddur (27: Ashrei p. 155):
עיני כל אליך ישברו עיניך על כל בריותיך בין אדם בין בהמה בין חיה בין עוף בין דגים בין שקצים בין רמשים...ומצפים לך, ועורגים אליך לתת למו אוכל...וכולם נושאים עיניהם לשמים לבקש אוכל
The eyes of all look to you for sustenance: your eyes are on all of your creations, whether human, domesticated animals, undomesticated animals, fowl, fish, or crawling creatures...And they hope for you and long for you give them food...And they all raise their eyes to heaven to ask for food. (Trans. my own).
Additionally, Psalms (147:9) states:
נוֹתֵן לִבְהֵמָה לַחְמָהּ; לִבְנֵי עֹרֵב, אֲשֶׁר יִקְרָאוּ
He giveth to the beast his food, and to the young ravens which call out.
R. Moshe Alshikh explains (there) that the calling out of the ravens refers to calling out to God.
Indeed this is nearly explicit in Job (38:41):
מִי יָכִין לָעֹרֵב, צֵידוֹ: כִּי-יְלָדָו, אֶל-אֵל יְשַׁוֵּעוּ; יִתְעוּ, לִבְלִי-אֹכֶל.
Who provideth for the raven his prey, when his young ones cry unto God, and wander for lack of food?
However, other commentators explain these verses differently. Regarding the first verse (Psalms 104:21), Radak and Malbim (there) explain that this does not mean that they literally pray to God.
In the words of Radak:
ולבקש מאל אכלם, כי הוא נותן לחם לכל בשר בהשתלשלות הסיבות, וכאילו הם מבקשים ממנו אוכלם בבקשם טרפם
And to request their food from God: For he provides food to all beings, through the chain of events, and it is as though they request their food from him, when they seek their prey.
Similarly, the Malbim there explains that when it says that they request their food from God, it means they seek out their food from those species that God has designed for their consumption.
In regards to the second verse (Psalms 145:15) as well, many commentators understand that "all those who look to God for sustenance" refers to humans; not animals. (Cf. Tol'dot Yitshak to Leviticus 25:3).
Regarding the third verse (Psalms 147:9), Radak (there) explains that the animals cry out "as though to God", but not actually to God.
Regarding the verse in Job (38:41), Rav Sa'adya Gaon gives the exact same explanation (in his commentary to the verse); that they cry out "as though to God".
R. Qafih explains (in his edition of Rav Sa'adya Gaon's commentary to Job there), that Rav Sa'adya Gaon inserts the words "as though" since animals are not intelligent enough to pray.
In summary, according to the understanding of the Rokeah, Alshikh, and Metsudat David, verses in Psalms indicate a certain connection the animals have with God, but according to the other commentators, these verses would be inconclusive. Indeed some such as Rav Sa'adya Gaon and Radak take pain to obviate this view.