I am unable to find any Punic myths concerning deities. Are there any such myths? If not, why not?
Unfortunately, the vast majority of Punic and indeed most Phoenician sources did not survive. The extant corpus is mostly inscriptions, typically shorter religious formulations, and consequently the greater part of their mythology has been lost to us.
But while there's little direct attestation of Carthaginian mythology, scholars believe some elements of Punic mythology were preserved in the writings of Graeco-Roman writers (for example, this 'fable' reported by Sallust). For the purposes of this question specifically, the legend of Queen Dido - which some scholars argue was originally a native Carthaginian founding myth - might be of interest:
Dido perhaps means 'the wanderer". It was her name among the Carthaginians, whereas at Tyre, the great Phoenician city which had founded Carthage or 'New Town' (allegedly in 814 BC), she was known was Elissa. This name was derived from a divinity, El. That is to say, Dido-Elissa was originally a goddess.
Additionally, Carthage was a Phoenician colony that maintained close ties with her mother city, Tyre, for centuries. As late as 332 BC, there were still Carthaginian envoys bringing tribute to Tyre's temple caught up in Alexander the Great's capture of the city. Many of Tyre's elderly, women, and children were also evacuated to Carthage prior to the siege. Even the word 'Punic' actually came from the Latin word for Phoenicians.
With such close cultural and demographical links, it can be reasonably inferred that most if not nearly all old Phoenician mythology were also carried over to Carthage. While that's still really not much, it does give us the fragmentary works of Sanchuniathon, which contain, among other stories, a creation myth and genealogies of gods including ones definitively worshipped in Carthage.
The overwhelming majority of our information about ancient Carthage comes from Greek and Roman sources, which do not document the myths.
If the Carthaginians documented their own myths, the documents did not survive. In fact, the vast majority of direct evidence for their religion derives from their use of theophoric names (invoking the god in a name) or in inscriptions, neither of which contain enough information to reconstruct any myths.
A 500 Piece Puzzle with 493 Pieces Missing!
Not a myth per se, but there is "Hanno's Prayer" in the Poenulus (Plautus). Act five supposedly contains his speech in Punic (or perhaps mock Punic? or perhaps garbled Punic? -- Igpayim Unicpayim?)
The translation at that link to Tufts is interesting, if legit. We do know that Plautus wrote the play some short while after the second Punic war, and that people would have been familiar with Carthaginians as non-Greek foreigners. If this contains any legitimate Punic words it might be of interest for you to look into.
I'm no Semiticist, but alonim vualonuth certainly looks like "gods and goddesses to me. I would take vu- to be and, and it does seem to look and behave like w- in Aramaic, from Proto Semitic wa.