The Gemara (Berachot 10b) records a debate between Bet Hillel and Bet Shammai regarding the words “Beshochbecha uvkumecha” (lit. “when you lie and when you arise”) and its application the the recital of Shema. Bet Shammai understands that Shema should be recited while in lying position at night, whereas in the morning, it should be recited while standing up. Bet Hillel, on the other hand, interprets the words to mean that the Shema should be recited at the time that people tend to lie down, that is, at night and also at the time when people arise, that is, in the morning. The accepted Halacha follows the opinion of Bet Hillel.
The Shulhan Aruch (O.H. 63:2) writes that if one wishes to be strict and specifically stand up while reciting Shema in the morning, he is considered to be a sinner, because he disregards the Halacha that the Sages accepted. Nevertheless, the Ben Ish Hai (Parashat Vaera, § 15) and the Mishna Berura (M.B., O.H. 63:5) say that if one wishes to stand because it will enhance one’s focus, then it is permissible. Similarly, if one stands up for Shema because it will prevent one from becoming tired in the prayer, for example, and one’s fellow congregants know that one is not doing it to follow the opinion of Bet Shammai, then it is also allowed.
The Shulhan Aruch (ibid:1) also says that although one may recite Shema while walking, while riding an animal or while reclining, one should not do so while lying flat on one’s back or on one’s stomach as it is appears irreverent. Therefore, if one is sick in bed and one wishes not to miss the time for Shema, one should turn to one’s side or at least raise one’s body to recite it. Similarly, if one is walking and wishes to recite Shema, the Shulhan Aruch (ibid:3) says that one should stop and recite at least the first verse while stationary. The Ben Ish Hai (ibid., § 14) and others say that one should be stringent and remain stationary until at least the words “Al Levavecha”. Furthermore, if one is at work or the like, one should stop what one is doing to recite Shema with the proper focus and decorum.
Summary: One may recite Shema in the morning while standing up as long as it not because one is specifically following the opinion of Bet Shammai. Although one may recite Shema in different positions, it is preferable to do so in a manner befitting this Mitzvah.