I have inlet to a reactor formed by equimolar CH4 and NH3, I want to find the volume of this gas mixture.

In flow processes like this, it has zero value to know the volume of a mixture of gases at the entrance of any equipment. What has value is the volumetric flow rate $\dot{V}$, as you then said, which has units of $\pu{m^3 s^-1}$.

First I wanted to calculate the sum of individual volumetric flow rates from V = m*rho. However, I could not find the rho values at 1 atm and 1000C for mentioned gases. My first question is that is it okay to continue with this conditions?

Volumetric flow rates are not additive in a flow stream where you have more than one component, such as this case where you have two gases. Even though the stream is made of $\pu{50\%}$ of methane and $\pu{50\%}$ of ammonia (in moles), you cannot do something like $\dot{V}_\ce{NH3} + \dot{V}_\ce{CH3}$. This is because you don't have two different densities in this stream, since it forms a homogeneous phase. Hence, it only has one physical property $\rho$ across its space that characterises it.

You can calculate the volumetric flow rate as follows $$ \dot{V} = vS = \frac{\dot{n}}{\rho} \tag{1} $$ If you have the velocity and the cross sectional area of the reactor inlet, use the first equality. I suspect that you don't have those, so you will need to use the second equality where $\dot{n}$ is the molar flow rate in $\pu{mol s^-1}$ and $\rho$ is the molar density and $\pu{mol m^-3}$.

Since you know the conditions at the inlet you just apply an equation of state and find $\color{blue}{\rho}$ \begin{align} p &= f(\rho,T,y_\ce{CH4},y_\ce{NH3}) \\ \pu{1 atm} &= f(\color{blue}{\rho},\pu{1000 °C}, 0.5, 0.5) \tag{2} \end{align} where the unknown is the density in blue. Once you have calculated, you can go back to Eq. (1) and calculate $\dot{V}$. As @Poutnik stated in the comment, just go for the ideal gas equation of state. This will be an excellent estimation, since the pressure is almost standard and the temperature is very high.

I have inlet to a reactor formed by equimolar CH4 and NH3, I want to find the volume of this gas mixture.

In flow processes like this, it has zero value to know the volume of a mixture of gases at the entrance of any equipment. What has value is the volumetric flow rate $\dot{V}$, as you then said, which has units of $\pu{m^3 s^-1}$.

First I wanted to calculate the sum of individual volumetric flow rates from V = m*rho. However, I could not find the rho values at 1 atm and 1000C for mentioned gases. My first question is that is it okay to continue with this conditions?

When you have more than one component, you calculate the volumetric flow rate (as one), not the individual ones. Matter in that stream forms a homogeneous gas phase, and as such, unique physical properties characterise it, for instance the density $\rho$. You may imagine that if you have a pipe in which $10$ different substance travel, it will be impossible to measure the individual volumes that each one occupy.

You can calculate the volumetric flow rate as follows $$ \dot{V} = vS = \frac{\dot{n}}{\rho} \tag{1} $$ If you have the velocity $v$ and the cross sectional area $S$ at the reactor inlet, use the first equality. I suspect that you don't have those, so you will need to use the second equality where $\dot{n}$ is the molar flow rate in $\pu{mol s^-1}$ and $\rho$ is the molar density and $\pu{mol m^-3}$.

Since you know the conditions at the inlet you just apply an equation of state and find $\color{blue}{\rho}$ \begin{align} p &= f(\rho,T,y_\ce{CH4},y_\ce{NH3}) \\ \pu{1 atm} &= f(\color{blue}{\rho},\pu{1000 °C}, 0.5, 0.5) \tag{2} \end{align} where the unknown variable is the density in blue. Once you have calculated, you can go back to Eq. (1) and calculate $\dot{V}$. As @Poutnik stated in the comment, just go for the ideal gas equation of state. This will be an excellent estimation, since the pressure is almost standard and the temperature is very high.