Define a lax monoidal category $(\mathcal{C}, \{\otimes_n\},\{\gamma_{(i,j)}\}, \{i_a\}) $ as follows:

$\mathcal{C}$ is a category.

For each $n \in \mathbb{N}$, we have a functor of weight $n$, $\otimes_n: \mathcal{C}^n\to \mathcal{C}$ called the $n$-fold tensor.

Since $\otimes_n$ is a functor $\mathcal{C}^n\to \mathcal{C}$, we can consider the composition $\otimes_n(\otimes_{j_1}, ..., \otimes_{j_n}):=\otimes_{(n,j)}$. This makes it a functor in $\sum_{i=1}^n j_i:=\ell$ objects. Then we we have a morphism of functors $\gamma_{(n,j)}: \otimes_{(n,j)} \to \otimes_\ell$ that is natural in each of the $\ell$ coordinates.

For each object $a$ of $\mathcal{C}$, we have a map $i_a: a \to (a)$ which is a natural transformation $Id\to \otimes_1$.

More notation: $\otimes_n(\otimes_{j_1}(\otimes_{k^1_1},...,\otimes_{k^1_{j_1}}), ..., \otimes_{j_n}(\otimes_{k^n_1},...,\otimes_{k^n_{j_n}})):=\otimes_{(n,j,k)}$.

(Note: The n,j,k are not actually integers. They're a nasty multi-index notation.) Let $\epsilon=\sum_{i=1}^n j_i$, $\delta_i=\sum_{r=1}^{j_i}k^i_{r}$, and $\lambda=\sum_{i=1}^n \delta_i$ We require further that the $\gamma$ make the following diagrams commute (for every multi index in our notation):

$$\begin{matrix}&&\gamma_{(n,j)}&&\\ &\otimes_{(n,j,k)}&\to&\otimes_{(\ell,k)}&\\ \otimes_{r=1}^n\gamma_{(j,k)}&\downarrow&&\downarrow&\gamma_{(\ell,k)}\\ &\otimes_{(n,\delta)}&\to& \otimes_\lambda&\\ &&\gamma_{(n,\delta)}&& \end{matrix}$$

Further, we require that the natural transformation $i$ makes the following diagram commute:

$$\begin{matrix} &&\otimes_n(i)\\ &\otimes_n &\to& \otimes_n\circ\otimes_1\\ id&\downarrow&&\downarrow & \gamma_{(n,1)}\\ &\otimes_n&\to&\otimes_n\\ &&id \end{matrix}$$

$$\begin{matrix} &&i\otimes_n\\ &\otimes_n &\to& \otimes_1\circ\otimes_n\\ id&\downarrow&&\downarrow & \gamma_{(1,n)}\\ &\otimes_n&\to&\otimes_n\\ &&id \end{matrix}$$

This defines a lax monoidal category. We call a lax monoidal category weak, or unbiased weak if all of the $\gamma$ and i are isomorphisms. (If they are equalities, then this is an unbiased strict monoidal category).

Now, there is a proliferation of diagrams, to be sure, but every diagram merely verifies the associativity or unit individually for all finite tensor products individually. It is conceptually simpler than the standard notion of a weak monidal category. It's the naive approach.

However, a priori, none of our maps are associative if we leave the diagrams as they are this is called a lax monoidal category, and if we turn the arrows around, it's called a colax monoidal category. Then an answer to your question is: yes. There are lax monoidal functors that preserve the lax monoidal structure, which is not a priori associative (it is lax associative, which means we can go in one direction).

Edit: To give credit where credit is due, this is covered in Tom Leinster's book "Higher Categories, Higher Operads" in chapter 3.

Define a lax monoidal category $(\mathcal{C}, \{\otimes_n\},\{\gamma_{(i,j)}\}, \{i_a\}) $ as follows:

$\mathcal{C}$ is a category.

For each $n \in \mathbb{N}$, we have a functor of weight $n$, $\otimes_n: \mathcal{C}^n\to \mathcal{C}$ called the $n$-fold tensor.

Since $\otimes_n$ is a functor $\mathcal{C}^n\to \mathcal{C}$, we can consider the composition $\otimes_n(\otimes_{j_1}, ..., \otimes_{j_n}):=\otimes_{(n,j)}$. This makes it a functor in $\sum_{i=1}^n j_i:=\ell$ objects. Then we we have a morphism of functors $\gamma_{(n,j)}: \otimes_{(n,j)} \to \otimes_\ell$ that is natural in each of the $\ell$ coordinates.

For each object $a$ of $\mathcal{C}$, we have a map $i_a: a \to (a)$ which is a natural transformation $Id\to \otimes_1$.

More notation: $\otimes_n(\otimes_{j_1}(\otimes_{k^1_1},...,\otimes_{k^1_{j_1}}), ..., \otimes_{j_n}(\otimes_{k^n_1},...,\otimes_{k^n_{j_n}})):=\otimes_{(n,j,k)}$.

(Note: The n,j,k are not actually integers. They're a nasty multi-index notation.) Let $\epsilon=\sum_{i=1}^n j_i$, $\delta_i=\sum_{r=1}^{j_i}k^i_{r}$, and $\lambda=\sum_{i=1}^n \delta_i$ We require further that the $\gamma$ make the following diagrams commute (for every multi index in our notation):

$$\begin{matrix}&&\gamma_{(n,j)}&&\\ &\otimes_{(n,j,k)}&\to&\otimes_{(\ell,k)}&\\ \otimes_{r=1}^n\gamma_{(j,k)}&\downarrow&&\downarrow&\gamma_{(\ell,k)}\\ &\otimes_{(n,\delta)}&\to& \otimes_\lambda&\\ &&\gamma_{(n,\delta)}&& \end{matrix}$$

Further, we require that the natural transformation $i$ makes the following diagram commute:

$$\begin{matrix} &&\otimes_n(i)\\ &\otimes_n &\to& \otimes_n\circ\otimes_1\\ id&\downarrow&&\downarrow & \gamma_{(n,1)}\\ &\otimes_n&\to&\otimes_n\\ &&id \end{matrix}$$

$$\begin{matrix} &&i\otimes_n\\ &\otimes_n &\to& \otimes_1\circ\otimes_n\\ id&\downarrow&&\downarrow & \gamma_{(1,n)}\\ &\otimes_n&\to&\otimes_n\\ &&id \end{matrix}$$

This defines a lax monoidal category. We call a lax monoidal category weak, or unbiased weak if all of the $\gamma$ and i are isomorphisms. (If they are equalities, then this is an unbiased strict monoidal category).

Now, there is a proliferation of diagrams, to be sure, but every diagram merely verifies the associativity or unit individually for all finite tensor products individually. It is conceptually simpler than the standard notion of a weak monidal category. It's the naive approach.

However, a priori, none of our maps are associative invertible. If we leave the diagrams as they are this is called a lax monoidal category, and if we turn the arrows around, it's called a colax monoidal category. Then an answer to your question is: yes. There are lax monoidal functors that preserve the lax monoidal structure, which is not a priori associative (it is lax associative, which means we can go in one direction).

Edit: To give credit where credit is due, this is covered in Tom Leinster's book "Higher Categories, Higher Operads" in chapter 3.