Below I need to find the optimal results of $y_i$ and $x_{ij}$, where $a_i$,$b_{ij}$ and $c_i$ are constant numbers. $x_{ij}$ and $y_i$ are binary variables, while $v_i$ and $z_i$ are allowed to be non-integers.
How can I deal with it?
\begin{align} \text{minimize} ~~& \sum_{i=1}^L [y_i z_i+(1-y_i)c_i] \\ \text{subject to} ~~& z_{i} \geq \sum_{j=1}^M x_{ij} b_{ij},~~i\in[1,L], \\ ~~& y_i a_i+(1-y_i)v_i\leq N_i,~~i\in[1,L],\\ ~~& v_i\geq jx_{ij},~~i\in[1,L],j\in[1,M],\\ ~& x_{ij}\in\{0,1\},~~i\in[1,L], j\in[1,M], \\ ~& y_{i}\in\{0,1\},~~i\in[1,L]\\ \end{align}
I assume that $z_i$ and $v_i$ are continuous variables, $N_i$ is a constant, and $b_{i,j}$ is a nonnegative constant. So you have two products $y_i z_i$ and $y_i v_i$ that you want to avoid. For the first one, introduce a variable $w_i \ge 0$ to replace $y_i z_i$ in the objective and enforce $y_i=1 \implies w_i \geq \sum_{j=1}^M x_{ij} b_{ij}$ with linear big-M constraint $$\sum_{j=1}^M x_{ij} b_{ij} - w_i \le M_i(1-y_i).$$ Here, $M_i=\sum_{j=1}^M b_{ij}$ is a good choice.
Now consider the other product $y_i v_i$. Your second constraint enforces $y_i=1 \implies a_i \le N_i$ and $y_i=0 \implies v_i \le N_i$. To enforce the first implication, just preprocess by fixing $y_i=0$ if $a_i > N_i$. To enforce the second implication, impose linear big-M constraint $$v_i - N_i \le (j - N_i) y_i.$$
Now drop your second constraint and use a mixed integer linear programming solver. You can also drop your first constraint if you don't care about the value of $z_i$. Alternatively, you can postprocess to set $z_i=\sum_{j=1}^M x_{ij} b_{ij}$.