Cross-lagged Pearson correlation in R I have a dataset where I have two recordings (sessions) of two different variables.
set.seed(123)
data <- data.table(
    id = rep(1:20, each = 2),
    session = rep(1:2, times = 20),
    var1 = sample(1:100, size = 40, replace = TRUE),
    var2 = sample(5:10,  size = 40, replace = TRUE)
)

If I want to know the correlation between two variables at the same time point, I can simply calculate a Pearsons correlation:
#Cross-sectional Pearson correlation
data[session == 1, cor.test(var1, var2)]

However, if I want to know the correlation between var1 and var2 at different time points, should I use a cross-lagged Pearson correlation? And if so, how do I determine the lag?
I imagine it would look like so:
#Cross-lagged Pearson correlation
library(testcorr)
cc.test(data[session == 1, var1], data[session == 2, var1], max.lag = 1)

Since there are only two session, and the time lag between the two is 1 session, I imagine the lag should be defined as 1?
How about when I want to correlate the value of var1 at session 1 with the value of var2 at session 2? Do I use the same cross-lagged correlation?
cc.test(data[session == 1, var1], data[session == 2, var2], max.lag = 1)

 A: If you only have two time points, the lag order can only be zero (no lag, thus contemporaneous relationship) or one.
While I have never used cc.test before, I wonder if it is applicable to your data in its current format. Your data is not stored as a time series. The rows do not increase in time; instead, they alternate between session=1 and session=2.
If you want to test $H_0\colon \text{Corr}(y_t,x_s)=0$ for some time points $t$ and $s$, you can run a linear regression of $y_t$ on $x_s$ and inspect whether the slope coefficient is different from zero. In your case that would be
model <- lm(data[session == 1, var1]~data[session == 2, var2])
summary(model)

For your data, I get
Coefficients:
                         Estimate Std. Error t value Pr(>|t|)   
(Intercept)                79.904     23.332   3.425  0.00302 **
data[session == 2, var2]   -4.316      3.026  -1.426  0.17090   

which means the relevant $p$-value is 0.17090, above the conventional significance levels of 0.05 or 0.01. Thus $H_0$ of zero correlation between var1 in session 1 and var2 in session 2 would not be rejected.
