Did Jesus read the Tanakh in Aramaic? Probably not.
As Claude Tresmontant observed,
"There were oral translations in Aramaic of the sacred books written
in Hebrew; they were called targumin. A translator in the synagogue
would read aloud, translating a passage from the Torah or one of the
prophets. But in the era before the destruction of the Temple,
putting these translations into writing was formally prohibited." (The
Hebrew Christ p. 5)
So the Tanakh in the local synagogues would generally be a document written in Hebrew, from which Aramaic oral translations were sometimes given.
As Steve noted in a prior post, there are various instances in the Gospels where an Old Testament passage is quoted, but the reading is found in the LXX and not in any known Hebrew text. This, combined with effective scholarship suggesting the widespread knowledge of Greek in first century Galilee (e.g. here & here), serves as evidence for the claim that Jesus at least sometimes taught in Greek.
It is also possible that some of the LXX passages found in the Gospels are Greek translations of scriptural teachings Jesus gave in Aramaic or Hebrew. For a helpful summary of arguments that Jesus spoke both Aramaic & Hebrew see Frank Luke's analysis on this site here.
How to lose a job as a translator in 5 minutes
As a translator, I have learned that when translating a statement well-known in the target language, you do not free-translate the passage. A good translator will refer to an already accepted translation. For example, if I translated "veni, vidi, vici" as "I arrived, I saw, I conquered", I would lose the confidence of my audience, because everybody knows it's supposed to be rendered "I came, I saw, I conquered."
It would be both natural and expected that, where Jesus quoted the Hebrew scriptures verbatim, an author writing in Greek would usually refer to the accepted translation of the passage, found in the LXX. When the scriptures were paraphrased, a free-translation may be more acceptable.
We should not rule out the likelihood that Jesus was familiar with the written & spoken Hebrew Tanakh, the written & spoken Greek Septuagint, and had heard oral renderings of the Tanakh in Aramaic.
Perhaps the plainest (and admittedly ever-so-slightly over-simplified) rendering of the trilingual nature of Jesus' world I have encountered is that Aramaic was the language of the home, Hebrew was the language of the synagogue, and Greek was the language of the marketplace.