Complexity class of efficient streaming algorithms Consider the class of problems $\mathsf{StreamL}$ which can be solved in logarithmic space reading the input in a single pass from left to right. In other words:

$L \in \mathsf{StreamL}$ if there exists a Turing machine $M$ which decides $L$, where:

There are two tapes, the read-only input tape and the working tape

$M$ moves only to the left on the input tape, and uses at most $O(\log n)$ space on the working tape.



Has this class been studied?

My assumption is that the answer is yes, but I'm not yet aware of a definition of the class in the literature.
Most literature on streaming algorithms that I am aware of considers the complexity of solving specific algorithmic problems, and does not tackle structural complexity i.e. defining classes such as the above and determining their relationships.
There is also a large body of work on communication complexity classes. In this domain there is a relevant class called $\mathsf{P}^{cc}$ (see Babai, Frankl, and Simon 1986: Complexity classes in communication
complexity theory), which contains functions of two variables $f(x,y)$ where they can be solved using a small amount of communication between $x$ and $y$. This is related to $\mathsf{StreamL}$ above (for functions of two variables, $\mathsf{StreamL}$ is contained in $\mathsf{P}^{cc}$), but the class above is not limited to functions of two variables and enforces a stricter computational requirement.
The obvious inclusions are $\mathsf{REG} \subseteq \mathsf{StreamL} \subseteq \mathsf{L}$, and no apparent inclusion either way between $\mathsf{StreamL}$ and $\mathsf{NC}^1$.
Along with my comment above (noting that not even AC0 is in "StreamL"), let me say that that this class has been studied before; you just need to know what they used to call it.

Search for "one-way logspace" and you will find plenty of references. (Typically, past work treats it as a reducibility concept.) The papers that those papers reference should cover what you need.