Is there a nonzero solution to this infinite system of congruences?

Let $u_n = a s_n + b t_n + c s_{n+1}$. The stronger claim is true: for large enough values of $n$,
the number $u_n$ will be exactly divisible by a fixed power of $2$ that doesn't depend on $n$.

Let $u_n = a s_n + b t_n + c s_{n+1}$ then (by induction) $$u_{n} = u_{n-1} + 2 u_{n-2} + 4 u_{n-3}.$$

The polynomial $x^3 - x^2 - 2 x - 4$ is irreducible and has three roots $\alpha_1$, $\alpha_2$, and $\alpha_3$ in $\overline{\mathbf{Q}}$. By the general theory of recurrence relations, $$u_n = A_1 \alpha^n_1 + A_2 \alpha^n_2 + A_3 \alpha^n_3$$ for constants $A_1$, $A_2$, $A_3$. Since $u_n \in \mathbf{Q}$, we may additionally deduce that $A_i$ lie in $\mathbf{Q}(\alpha_1,\alpha_2,\alpha_3)$. That is because we can solve for $A_i$ using the equation

$$\left( \begin{matrix} \alpha_1 & \alpha_2 & \alpha_3 \\ \alpha^2_1 & \alpha^2_2 & \alpha^2_3 \\ \alpha^3_1 & \alpha^3_2 & \alpha^3_3 \end{matrix} \right) \left( \begin{matrix} A_1 \\ A_2 \\ A_3 \end{matrix} \right) = \left( \begin{matrix} u_1 \\ u_2 \\ u_3 \end{matrix} \right)$$ and the matrix on the left is invertible (Vandermonde). In fact we deduce the stronger claim that any Galois automorphism sending $\alpha_i$ to $\alpha_j$ sends $A_i$ to $A_j$. (Simply consider the action of the Galois group on both sides of this equation, noting that the $A_i$ are determined uniquely from this equation.) In particular, if one of the $A_i = 0$, then all of the $A_i = 0$.

But now fix an embedding of $\overline{\mathbf{Q}}$ into $\overline{\mathbf{Q}}_2$. From the Newton Polygon, we see that there is one root (call it $\alpha_1$) of valuation $0$, and the other two roots have valuation $1$. Hence $$\|A_1 \alpha^n_1 \|_2 = \|A_1\|_2, \quad \|A_2 \alpha^n_2 \|_2 = \|A_2\|_2 \cdot 2^{-n}, \quad \|A_3 \alpha^n_3 \|_2 = \|A_3\|_2 \cdot 2^{-n}.$$ In particular, if $A_1 \ne 0$, then (by the ultrametric inequality) $\| u_n \|_2 = \|A_1\|$ for $n$ large enough. Hence we deduce that either the $2$-adic valuation of $u_n$ is eventually constant (as claimed) or that $A_1 = 0$ and so $A_i = 0$ for all $i$, which implies that $u_n = 0$ for all $n$. But if $u_1 = u_2 = u_3 = 0$, then

$$\left( \begin{matrix} 1 &1 & 1 \\ 1 &3 & 3 \\ 3 & 5 & 9 \end{matrix} \right) \left( \begin{matrix} a \\ b \\ c \end{matrix} \right) = \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right)$$

The matrix on the left is invertible which implies that $a=b=c=0$.

$u_n=s_na + t_nb + s_{n+1}c$ satisfies the same recurrence relation as $s_n$ and $t_n$: $u_n = u_{n-1} +2u_{n-2} + 4u_{n-3}$. The question is whether $2^{n+1}\mid u_n$.

Since $v_n=u_n/2^{n+1}$ satisfies

$v_n = \displaystyle\frac{v_{n-1} +v_{n-2} + v_{n-3}}{2}$

the answer is affirmative only if there are $v_0, v_1, v_2$ (not all 0) such that $v_n$ is always integral.

EDIT. As sharply noticed by the OP, the attempt below was wrong, since a matrix I claimed to be invertible (mod $2$) is in fact singular. A similar, more computational argument does work (mod $5$).

It's clear that (mod $2$) such a sequence $v_n$ must follow either one of the 3-periodic patterns $000$ and $110$ (up to shifts). In the $000$ case keep dividing entire sequence $(v_n)$ by $2$ until one term is odd, and then shift the sequence to start with that term, so it's back to the $110$ case. Therefore it must be that $$\require{cancel}\cancel{\det\left (\begin{smallmatrix}v_1 & v_2 & v_3\\ v_2 & v_3 & v_4\\ v_3 & v_4 & v_5 \end{smallmatrix}\right )\equiv \det\left (\begin{smallmatrix}1 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 1 \end{smallmatrix}\right )\!\!\!\!\pmod{2} \ne0}$$

CORRECTED ARGUMENT. Notice that

$$D=\det\left (\begin{matrix}v_1 & v_2 & v_3\\ v_2 & v_3 & v_4\\ v_3 & v_4 & v_5 \end{matrix}\right )=\det\left (\begin{matrix}v_1 & v_2 & v_3\\ v_2 & v_3 & \frac{v_1+v_2+v_3}{2}\\ v_3 & \frac{v_1+v_2+v_3}{2} & \frac{v_1+3v_2+3v_3}{4} \end{matrix}\right )\\= \frac{-4v_3^3+4v_2 v_3^2+2v_1 v_3^2+v_2^2 v_3+5v_1 v_2 v_3-v_1^2 v_3-3v_2^3-2v_1 v_2^2-2v_1^2 v_2-v_1^3}{4}$$

is $\equiv 0 \pmod{5}$ if and only if $v_1\equiv v_2\equiv v_3\equiv 0 \pmod{5}$. This is proved by the following snippet of code:

awk -vp=5 'BEGIN {
    for(a=0; a<p; a++)
        for(b=0; b<p; b++)
            for(c=0; c<p; c++) {
                d=4*c^3-4*b*c^2-2*a*c^2-b^2*c-5*a*b*c+a^2*c+3*b^3+2*a*b^2+2*a^2*b+a^3;
                if(d%p==0) print a, b, c;
            }
}'

Now divide the entire sequence $(v_n)$ by a power of $5$ so that at least one term is is not $\equiv 0$, and shift it to start with that term, thus $v_1\not\equiv 0$ and therefore $D\ne0$.

Next this implies that there is an integral linear combination $(z_n)$ of $(v_n)$ and its shifts $(v_{n+1})$, $(v_{n+2})$ such that $z_1=z_2=0$, $z_3\ne 0$, and still $z_n=(z_{n-1} +z_{n-2} + z_{n-3})/2$ holds.

Finally write $z_3=2^m d$, with $d$ odd, and start the recursion from $0, 0, 2^m d$ to easily notice that it runs into a half-integer in $m+1$ steps.