I'll migrate my answer from the comments and add a bit.
Overall, I do believe that it's totally down to the individual how they write their name, especially so in Japanese. On the whole, foreigners, even overseas Japanese, very often have katakana attached to their names rather than kanji, even if they do have a version of their name that uses kanji -- the two examples that come to mind are Yoko Ono (known as オノ・ヨーコ rather than 小野洋子) and Michio Kaku (ミチオ・カク rather than 加來道雄 -- he even gets his name put in the Western order)
Using a name in Kanji will help a lot to seem less foreign, if that's your intention, which is why a fair number of naturalized Japanese and foreigners doing business in Japan will give themselves Japanese names using kanji (plus the law requires a name to be hiragana, katakana, or kanji). Kanji come off as more refined, too.
There is a contradiction to this though, in that Katakana surnames with Kanji forenames are nearly unheard of. I've only ever seen this once, and the fact that I remember it for being strange is a credit to how rare it really is.
For you, you could do a few things, and pick whichever suits you best:
Firstly, you could ask your parents what they thought Shiori meant (if anything). I know a lot of parents who give children foreign names at least make an effort to check they know what it means (or what they think it means), so this could work for you, and then you can deduce which kanji your parents were referencing.
If they didn't have anything in particular in mind, you could always pick one of the kanji options for Shiori yourself. Or, if you're a girl, it's okay to write your name in hiragana as しおり, which is commonplace in Japan amongst girls, but not boys. Because of this, it's seen as a very feminine trait.
Finally, you could Japanize your surname and pick a kanji for your name. This is probably the most Japanese option (As mentioned in the comments, Marutei Tsurunen did this, which is why I disagree strongly with the notion that Kanji isn't for foreigners), or otherwise just make up a new surname to use in Japanese. But if you're using a katakana surname, a katakana forename is the only natural option, as I mentioned earlier.
Don't feel pressurized to pick a Kanji unless it suits you. There are a lot of native Japanese with hiragana names, and lots of Korean Zainichi who even just use kana completely (even though most Koreans can write their names in 漢字). Choose what type of impression you want to give off with your name, and whether you want to feel more Japanese with your name (while being mindful that they'll be able to tell in reality if you're not ;), more feminine (if you're a girl), whether katakana's fine for your tastes and so on. I use a Kanji name as most Japanese do, except in my case it's (by virtue of my name) clearly a Chinese name.
Figure out what type of a name you think suits your personality and who you are, but don't get too caught up on an option fitting 100% perfectly. As Shakespeare said, a rose by any other name would smell just as sweet.