First let's show that for any integer $n$, $n^4\equiv 0, 1\pmod{16}$. 

• If $n\equiv 0, \pm 2, \pm 4, \pm 6, 8\pmod{16}$, we have $n^4\equiv 0\pmod{16}$. 
• If $n\equiv \pm 1, \pm 3, \pm 5, \pm 7\pmod{16}$, we have $n^4\equiv 1\pmod{16}$.

Therefore, the value of $(x_1^4 + x_2^4 + \cdots + x_{14}^4)\pmod{16}$ can be $0$, $1$, $\cdots$, $14$, but will never be $15$. However $9999\equiv 15\pmod{16}$. This means that the given equation is insolvable.