One of the most famous debates in Jewish history was over the issue of saying the poem מכניסי רחמים where we supplicate the angels and ask them to bring our prayers before Hashem [and other similar prayers where we turn to the angels for help]. The problem is that the Rambam [Hakdama to Chelek in the fifth of the 13 foundations of our faith] rules that one may not daven to anyone other than Hashem, so what business do we have turning to the angels as intermediaries? If you need to speak to a big politician, or li-havdil a famous rabbi, sometimes you have to go through intermediaries but Hashem is accessible to all and it is FORBIDDEN to turn to anyone else.
In fact, the Maharal [Nesivos Olam chapter 12] changed the language of the tfilla so it won't sound like we are asking the angels for help [יכניסו רחמינו instead of הכניסו רחמינו]. The Chasam Sofer would say a long tachanun while everybody else was saying the poem so he wouldn't have to join. Many others as well avoided saying it. However the custom in most communities is to say it and it is printed in all of the slichos books. There is a mabul of literature on the topic and I will not quote all of the sources I know about that discuss it. That would make me seem more scholarly and knowledgable than I am. The one source I will note is the Yeshurun Journal in Volume 3 where there is an excellent, thorough treatment of the topic.
I will add something interesting which I didn't see anyone bring up: In some editions of the Rambam he writes וכן אין ראוי לעבדם כדי להיותם אמצעים לקרבם אליו אלא אליו בלבד יכוונו המחשבות ויניחו כל מה שזולתו
Meaning that one is not allowed to SERVE anyone in order that he should be an intermediary between us and Hashem, rather all of our thoughts should be directed at Hashem Himself. This would indicate that if one is not actually SERVING the angels then just asking them for help would be fine and thus מכניסי רחמים would be permitted. [This was pointed out by Rav Asher Weiss Shlita in Minchas Asher on the Moadim סימן א].
However, in the Rav Kapich edition of the Rambam and so in the Rav Shilat edition the Rambam's words are rendered [from the original Arabic] as ואין עושין אותם אמצעים להגיע בהם אליו אלא כלפיו יתעלה יכוונו המחשבות ויניחו כל מה שזולתו. Meaning that one is not allowed to use the angels as intermediaries to get to Hashem. According to this the Rambam omits any mention of serving the angels and simply forbids turning to them at all as a means of getting closer to Hashem. This would effectively forbid us from saying מכניסי רחמים because that is exactly what we do.
So whether it is permitted or not would be dependent on what is the more accurate translation of the Rambam's words.
