The answer is no, but it's almost yes. A stochastic Turing machine can find a diagonally non-computable ($\textsf{DNC}$) function ($f$ with $f(x)\ne\varphi_x(x)$ for all $x$) and finding a complete extension of PA is equivalent to finding a $\textsf{DNC}$ function with $f(x)\in\{0,1\}$ for all $x$.

Frank Stephan showed that a stochastic TM cannot find any completion of PA. (The fact that it cannot find any given one, as in Carl Mummert's answer, was known earlier.)

The answer is no, but it's almost yes. A stochastic Turing machine can find a diagonally non-computable ($\textsf{DNC}$) function ($f$ with $f(x)\ne\varphi_x(x)$ for all $x$) and finding a complete extension of PA is equivalent to finding a $\textsf{DNC}$ function with $f(x)\in\{0,1\}$ for all $x$.

Antonin Kucera, in Measure, $\Pi^0_1$ classes, and complete extensions of PA, in Recursion Theory Week, Lecture Notes in Mathematics 1141, 245-259, showed that a stochastic TM cannot find any completion of PA. (The fact that it cannot find any given one, as in Carl Mummert's answer, was known earlier.)