Traditionally, people did not wear tallit and tefillin during Selichot, for the simple reason that the appropriate times for Selichot are after midnight or very early in the morning,[1] at both of which one is not supposed to put on his tallit[2] and tefillin.[3] Therefore, wearing tallit and tefillin could not be an integral part of the institution of Selichot.[4] This logically remains the case even when Selichot are recited when there is enough daylight to put them on. It is therefore not proper to miss some of the Selichot because one was putting tallit and tefillin on before starting Selichot.
However, it is a good question whether there is a positive effect for one to wear tallit and tefillin for Selichot when possible. Selichot are a special type of tefilla[5] that overlaps with elements of our daily tefilla (especially for those who say Yud Gimmel Middot daily). Thus, our question essentially depends on whether tallit are tefillin enhance tefilla in general. Let us consider tallit and tefillin individually.
We have discussed elsewhere[6] the reasons that married men wear a tallit at Shacharit even when they are already wearing tzitzit (as well as why these reasons may not be sufficiently important to compel single men do so[7]). One approach we cited is that it is sensible to make sure that one is wearing valid tzitzit (our “tzitzit” garment might be too small), which has special significance at Shacharit, when we recite the Torah section on tzitzit.[8] According to this reasoning, Selichot, in which we do not mention tzitzit, is not a time when a tallit is called for. Another possibility is that a tallit enables one to cover his head, which helps improve his concentration. Consequently, perhaps one could argue that there is a preference for one to cover his head with a tallit during Selichot.[9] On the other hand, why is this more important at Selichot than at Mincha, at which we do not wear a tallit? A possible rejoinder is that since one is going to wear the tallit for the subsequent Shacharit, he might as well put it on prior to Selichot.
The Taz[10] discusses the minhag that the chazan for Selichot wears a tallit, noting the proper way for him to do so without needing the corresponding beracha, which one is not allowed to say at night. The Taz argues that since we recite the Yud Gimmel Middot at Selichot, there is cause for the chazan to be properly cloaked, especially in light of the gemara[11] that states that Moshe saw HaShem wrapped up like a chazan when He taught Moshe how to recite the Yud Gimmel Middot. Other authorities discuss whether this minhag is advisable, considering the kabbalistic reasons not to put on tzitzit at night.[12] All the discussants seem to assume that only the chazan wears a tallit for Selichot. One could argue, however, that the reason that these classical poskim assumed that most people were not wearing a tallit is because in their time, Selichot were recited at night, when wearing a tallit poses a problem. If so, in the case of contemporary morning Selichot, all participants might be encouraged to wear one.
Shacharit is the chosen time for tefillin because we need to wear them during the day, in a state of cleanliness and pure thought, and because they are mentioned in Kri’at Shema.[13] There is a connection, although a weak one, between tefillin and tefilla.[14] While one might presume that the connection should apply to Selichot as well, we again note that the connection to Selichot does not appear to be more important than the connection to Mincha. (There is an opinion that it is proper to don tefillin at Mincha as well,[15] but that is clearly not a common practice). Again, it is possible to make a case that if one is putting tefillin on soon anyway, he might as well put them on for Selichot (as opposed to Mincha, when one would have to bring his tefillin specifically for the prayers).
In summary, we have seen that it is not important to wear tallit and tefillin during Selichot. However, we raised the possibility, without successfully confirming or contradicting it, that there is some value in putting them on before Selichot. Therefore, no matter how it works out practically for you (including time and concentration considerations), putting on the tallit and tefillin either before or after Selichot is fine.