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In a famous Midrash in Shab. 87a-b, the angels object to the Torah being given to humans, until Moshe proves that it is only relevant to humans because of their nature. In one example, he asks rhetorically, "Do you (angels) have jealousy (that might lead you to kill or covet)?"
Well, to be honest, they were acting rather jealous. So, do angels have jealousy?
It is clear that angels have jealousy towards humans from Tosfiyos Brachos 3a that says that some say that we say certain prayers in Aramaic in order that the angels should not be jealous of us, and Tosfiyos does not say there is no jealousy, only that we say other prayers in Lashon Kodesh so that can not be the reason. Also Rashi Braishis 1:26 indicates that angels have jealousy towards humans.
Per Sifsei Chachamim Breishis 1:26 the angels do not have jealousy among themselves, however they do have jealousy towards human beings. (Thanks to IsraelReader for the source).
Berachos 16b-17a implies that angels do fight with each other (see top Rashi on 17a), and Rabbeinu Bechaye (Bereishis 18:2) writes that the posuk that we say in davening עושה שלום במרומיו (Job 25) means that Hashem gives distinct tasks to each angel so that they dont end up encroaching upon each other's roles and fighting. So we see that angels arent really flawless.
I once heard Rabbi Shafier explain that angels have free will but dont sin because they see Hashem so clearly that for them, the choice to sin would be idiotic, comparable to a man sticking his hand in a fiery furnace to win a bet. Does he have the free choice to do so? Certainly, we have the free choice to injure ourselves, but it is an easy choice not to make. So too with angels: jealousy is something we feel when are insecure if what we have is best suited for us, or perhaps if the possessions of someone else more appropriately belong to me! Ibn Ezra on Lo sachmod (Yisro) explains that the way to avoid feeling jealous is to recognize that Hashem runs the world and knows what He is doing, and therefore I really do have the wife/car/house/life that is best for me. But man has a difficult time keeping this is mind and so he comes to murder, theft, and even adultery. But what Moshe was telling the angels was that mal'achim see Hashem so clearly that they have no reason to be envious of each other to the extent of murder and robbery.
However when it came to Hashem's decision to give the Torah to mankind, the angels were less understanding of why Hashem wanted to give His special Laws to imperfect man. Without the clear understanding of Hashem's ways that they usually have, they were as jealous as any man generally is.
Gavriel is sent to destroy Jerusalem, Hashem commands him to fill his hands with fiery coals from the hands of the Cherub. Gavriel goes to the Cherub, (understood to be Michael), Explains what he's supposed to do, Michael says, 'take the coals', Gavriel says'put them in my hands', directly disobeying G-d, with consequence. He is cast out from behind the curtain and lashed 60 times, and replaced by Dubiel, the angel of Perisa, for 21 days. (yoma 77a, tanchuma tazria 9, see also Malachei Elyon by Reuven Margoleous).
My friend said in the name of his rabbi that angels are preprogrammed to act in certain ways, regardless of whether or not those ways are right or in accordance with Hashem's will. I asked my rebbi, Rav Aron Lopiansky what angels are, he said that 'they are middos of HKBH that sometimes take on minds of their own'. So they disobey Him sometimes, because that is how they were made to act.
So the angels got upset that Moshe was taking the torah because they were created to act that way at that time, probably to teach us something, as the story is related in the Talmud.
The envy of the Malachom had to for Klal Yisroel receiving the Torah was not what Moshe was referring to when he spoke about jealousy. Envying someone for doing something good is not the prohibition of Lo Sacmod
The Gemara (Bava Basra 21a) says that although kinah – envy – is generally a destructive and evil trait, sometimes it can be quite positive and useful. It brings the posuk in Mishely. “Kinas sofrim tarbeh chochmah – the [proper] envy of scholars stimulates more learning.”
Angels probably do have the positive type of jealousy.
The Be'er Basadeh on Bereishis 1:26 rejects the answer of the Sifsei Chachamim quoted by @gershongold above that they can't be jealous of each other but they could be jealous of humans because it makes no sense. Angels are entirely holy with no element of jealousy or hatred. They only serve Hashem with no externalities, just a fear of Hashem.
Instead, the Be'er Basadeh suggests that the jealousy we find them having throughout Chazal is only a jealousy on Hashem's behalf similar to what we find by Pinchas in Bamidbar 25:11:
בקנאו את־קנאתי
"When he was jealous with my jealousy"
