I don't believe that there should be mediators (like Christians believe Jesus is a mediator between God and man whom they pray to and/or pray through) between us and God, but is there anywhere in the Hebrew Bible where this subject is alluded to?
There are levels in our connection with Hashem.
The prophets during all ages acted as such. Though - all of Israel are encouraged to reach that level (of prophecy) themselves. Still, when Hashem wanted to say something, it was through prophets. In the Torah we find it the other way too - Am Yisrael asks Moshe to speak to Hashem for them.
There is an example of a non-prophet speaking directly to Hashem. In Shmuel, however, we see Channa praying - speaking to Hashem herself. Still, her answer came from Eli - a prophet, not directly to her.
Kohanim as well, were assigned to the work in the mikdash. Holy work bringing us closer to Hashem - but only a kohen could be the one who brings a sacrifice to the mizbeach.
And, of course, the kohen gadol is the only one who can do the Yom Kippur avoda for us.
I once typed up something about angels in prayer, so I'll use it here, since it's very relevant (I've translated the Hebrew):
Maharal, Nesivos Olam, Nesiv HaAvodah ch. 12: ... This implies that he should pray to G-d that he not have any distressing occurrences. However, to pray "Angels, don't distress me," is not proper, unless it's done in the manner of prayer to G-d. But we have the custom to say [in Slichos] "[Angels] who bring our prayers of mercy [to G-d], bring in our prayers of mercy before the master of mercy"* etc., and it isn't proper, because it's like praying to angels to bring prayers of mercy before G-d, and we find no source validating this. ... However, there is room to defend [the prayer], that when we say "[Angels] who bring our prayers for mercy" etc. is not at all a request, just that the person is saying to the angels who bring in prayers of mercy to bring the prayer for mercy to G-d, and a person has the power to command angels to do this [and he is commanding, not requesting from them, and therefore there is no problem because he isn't praying to angels]. Nevertheless, it is not proper to say "[Angels] who bring in our prayers for mercy" as anything other than a prayer, like our rabbis said that a person should pray that everyone strengthen him and that he not have enemies or distressors and pray to G-d concerning this [which strongly suggests this prayer is meant as a request to the angels]. Therefore, what he requests in "[Angels] who bring in our prayers for mercy" is a prayer to G-d that the angels who bring in prayers for mercy should bring in the prayers for mercy. According to this, it is more correct to edit the prayer to say "[Angels] who bring in our prayers for mercy will bring in our prayer for mercy, those who cause prayer to be heard will make our prayer heard, those who bring in tears will bring in tears." ... But to say "[Angels] who bring in prayers for mercy, bring in our prayers for mercy mercy" in prayer shouldn't be said at all; only "[Angels] who bring in prayers for mercy will bring in our prayers for mercy," etc.
*The full text is "Angels who bring in prayers for mercy, bring in our prayers for mercy in front of the master of mercy. Angels who cause prayers to be heard, bring in our prayers before ... and so on.
Shu"t Chasam Sofer [1:162] (which I quoted also in this answer: I am surprised at the genius Maharal MiPrag [above] in the work Nesivos Olam, Nesiv HaAvodah ch. 12, who refused to say the prayer "[Angels] who bring in prayers for mercy," because we have nothing with the angels, only G-d, Who hears the voice of our prayers: ... All the books are full of "And she went to seek out G-d," to pray, like the Rambam says on the Torah [?]. There is no need to speak at length about how surprised I am, because that same genius writes this in the same chapter at a distance of the amount of time it takes to say a few words! Since his words are slightly hard to understand, I will therefore explain it this way: It's the ordinary way to have an intermediary between a king and an ordinary person when the ordinary person isn't important compared to the king, or if the ordinary person can't express his words properly. Since Israel don't need an interpreter between them and G-d Who loves them, and He accepts graciously even gibberish from them, then the intermediary between them displays only that they have little faith, G-d forbid. ... According to the words of the aforementioned genius, we should also have to omit the prayers "Angels of mercy" and the slichah that contains the thirteen attributes [i.e. "ezerah elokim ve'ehemayah..."].
Pischei Teshuvah (Isserlein) to Orach Chayim 581:1: In the responsa Shemesh Tzedakah (#23-24) he brings the language of Rashi in Sotah that an individual requires the help of the angels, which implies that there is no prohibition involved [in requesting help from them]. ... And in the responsa of Mahari Bruna #275 he says that we have no reason to have any negative feelings at all about what we say "[Angels] who bring in prayers" and "Angels of mercy," because this isn't a request in the way of an intermediary, only humility, that he is embarrassed to approach the king [G-d].