It depends on the distribution. For instance, if we restrict our attention to the case where $\lambda_{ij}$ is uniform on $[0, \bar{\lambda}_{ij}]$, then we can choose $\bar{\lambda}_{ij}$ so as to make $T$ a.s. injective, but we can also chose them so that this is not the case.

By taking transposes, it's equivalent to ask whether $T$ has dense image.

For one direction, let's begin by seeing how to choose $\bar{\lambda}_{ij}$ so that $e_1$ is almost surely in the closure of the image. Let $a_j$ be your favorite square-summable sequence of positive numbers, and set $\bar{\lambda}_{1j} = a_j$. Then select $\bar{\lambda}_{ij}$, $i \ne 1$, so that for each $j$ we have $\sum_{i=2}^\infty |\bar{\lambda}_{ij}|^2 < \frac{1}{j} a_j^2$. Generate a matrix $T$ with the resulting distribution. Almost surely, we have $\lambda_{1j} \le \frac{1}{2} a_j$ for infinitely many values of $j$. For such a $j$, we have $T(\lambda_{1j}^{-1} e_j) = e_1 + \lambda_{1j}^{-1} \sum_{i=1}^\infty \lambda_{ij} e_i$, so
$$\begin{align*}\|T(\lambda_{1j}^{-1} e_j) - e_1\|^2 &= \lambda_{1j}^{-2} \sum_{i=2}^\infty |\bar{\lambda}_{ij}|^2\\ &\le \frac{1}{j} \lambda_{1j}^{-2} a_j^2 \\ &\le \frac{4}{j}\end{align*}$$
Thus $e_1$ is in the closure of the image.

Now you can construct similar matrices for which $e_2, e_3, \dots$ are in the closure of the image, and shuffle together their columns so that the image of the resulting matrix contains the union of their images, and thus is dense.

In the other direction, choose $a_j$ as before and set $\bar{\lambda}_{j+1, j} = a_j$. This matrix is going to look somewhat like a right shift. Choose another sequence $c_j$ so small that $\prod_{j=1}^\infty (1 - 3c_j) > 0$, and choose $\bar{\lambda}_{ij}$, $i \ne j+1$, so that $\sum_{j=1}^\infty (c_j a_j)^{-1} \sum_{i \ne j+1} |\bar{\lambda}_{ij}|^2 \le 1$. Now by splitting up the matrix $T$ into the entries on and off the shifted diagonal, we can write
$$T = SUA + MCA$$ where $$\begin{align*}A &= \operatorname{diag}(a_1, a_2, \dots) \\
C &= \operatorname{diag}(c_1, c_2, \dots) \\S &= \text{right shift} \\ U &= \operatorname{diag}(U_1, U_2, \dots), && U_j \text{ iid } \mathrm{unif}(0,1) \\
\|M\| &\le \|M\|_{HS}\le 1. \end{align*}$$
The $c_j$ were chosen so that with positive probability, we have $U_j \ge 3c_j$ for all $j$. On this event, for any vector $x$ we have $\|Ux\|^2 = \sum_j |U_j x_j|^2 \ge \sum_j |3 c_j x_j|^2 = (3 \|Cx\|)^2$. Since $S$ is an isometry, we have $\|SUAx\| = \|UAx\| \ge 3 \|CAx\|$. And $\|MCAx\| \le \|CAx\|$ since $\|M\| \le 1$. Hence $\|Tx\| \ge \|SUAx\| - \|MCAx\| \ge 2 \|CAx\|$ for any $x$.

On the other hand, we have $$|\langle Tx, e_1 \rangle| = |\langle MCAx, e_1\rangle| \le \|MCAx\| \le \|CAx\|.$$ Combining these gives $|\langle Tx, e_1 \rangle| \le \frac{1}{2} \|Tx\|$ for all $x$. In particular, whenever $\|Tx \| = 1$ we have $\|Tx - e_1\|^2 = 2 - 2 \langle Tx, e_1 \rangle \ge 1$. So $e_1$ is not in the closure of the range of $T$.

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