The problem with that is that Wood Release is a Kekkei Genkai of ether the Senju Clan (or what ever clan Hashirama mother was from) and it was stated by Kishi himself that Sakura has no special background.
It's more of a Hashirama Senju exclusive ability than a Senju exclusive ability. Unless it's revealed very soon that Hashi did inherit his Wood Style from one or both of his parents as a matter of fact, I doubt that that's the case here. It seems to me that Hashi was the only one in his clan to posses Wood Release Techniques so I doubt that it's exclusive to only the Senju clan. But you may ask why then would Madara go for Hashi's DNA if other people have the potential to posses it? Here's my explanation:
1. Hashi being a Senju is a decedent of the Six Path Sage's younger son, so Madara obviously went for the one with the DNA that was necessary for him to unlock the ability to use the Rinnegan.
2. If his Wood Release was a factor, as far as he was concerned, Hashi was the only Wood Release user that Madara knew of at the time.
3. If someone like say, Sakura (hypothetically speaking of course), could use Wood Release, a possible explanation for why Danzo, Orochimaru, or anyone else interested in getting Wood Release via artificial means would be because of this. Since Hashi comes from a clan who are basically the descendants of the second born son of the Sage of the Six Paths, then Hashi's Wood Release might be a more special case. If you factor in Hashi's background and the vitality of his Wood Release, one could conclude that Hashi's Wood Release has a level of vitality on par with the Sage's Younger Son due to the Son inheriting the "body" of the Sage, thus setting it apart from other potential Wood Release users should Kishi decide to reveal characters not related to Hashi having that same ability.
4. Another reason I feel that Sakura having Wood Release without having to have a Senju background could be potentially possible would be because of these two points of interest: Magnet Release and Lava Release. The users of Magnet Release are Toroi, the Third Kazekage and Gaara's Father (AKA the Fourth Kazekage). For Lava Release, the users of this advanced nature are Dodai, Kurotsuchi, Roshi (via his Tailed Beast named Son Goku) and Mei Terumi (AKA the Fifth Mizukage). The users of Magnet Release come from either the Sand Village or Cloud Village while the users of Lava Release come from either the Cloud Village, the Rock Village or the Mist Village. The one thing that all these characters have in common is that none of them appear to be related by blood in any sense. This leaves me to believe that while any Advanced Nature is still a Kekkei Genkai by default, it isn't one in the traditional sense that only members of a certain clan can perform this certain technique. Rather, an Advanced Nature can be found in multiple clans and non-clans alike throughout many unrelated lineages alike. It's possible that advanced nature genetics are more widespread in the gene pool of the Narutoverse then the genetics of more tight-nit clans such as the Hyuga Clan or Uchiha Clan, unless a certain clan actually goes out of it's way to keep it's advanced nature within the gene pool of the clan itself, as it seems to be the case with the Yuki Clan.
5. Another thing to note is that the Magnet Release users had differing methods of channeling their powers (Toroi with his custom shuriken, the Third Kazekage with his Iron Sand and Gaara's Father with his Gold Dust) as well as the Lava Release users differing forms of lava (Mei and Roshi's molten rock variety, Kurotsuchi's quicklime variety and Dodai's vulcanized rubber variety). If these differences in appearance despite being the same element are to indicate anything, then this is how I see it. If Sakura where to reveal the ability to perform Wood Release without having the need for Senju DNA, then maybe instead of just growing trees with green leaves on them, perhaps her variety of Wood Release would have her specialize in techniques that involve growing Cherry Blossom Trees so that it would fit in with her theme of Cherry Blossoms.
Welp, that took forever to type, but it was worth it.