The variance-inflation factor for an individual predictor in an ordinary least-squares model is based on the multiple $R^2$ for its linear regression against all the other predictors.* Any correlation with the other predictors inflates the variance of its coefficient estimate beyond an uncorrelated situation.
The quadratic form involved in a simultaneous "chunk" test on multiple coefficients, in contrast, means that correlations between predictors can either increase or decrease the value of the test statistic depending on the signs of correlations and of coefficient estimates.
A simple example with two predictors illustrates the principle, with the additional simplification that both predictors have the same variance inflation factor. Start with the usual ordinary least squares setup, with continuous outcome $y$ and normally distributed error $\epsilon$:
$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \epsilon.$$
Center and scale each of $x_1$ and $x_2$ to zero mean and unit standard deviation. That simplifies the design matrix $X$ so that:
$$X^TX=N \begin{pmatrix}1 & 0 &0\\0&1&r\\0& r& 1 \end{pmatrix}$$
where $N$ is the number of observations and $r$ is the sample correlation coefficient between $x_1$ and $x_2$. The covariance matrix of coefficient estimates, with $\hat\sigma^2$ the estimate of error variance, is:
$$\hat\sigma^2 (X^TX)^{-1} = \frac{\hat\sigma^2}{N} \begin{pmatrix}1 & 0 &0\\0&\frac{1}{1-r^2}&\frac{-r}{1-r^2}\\0& \frac{-r}{1-r^2}& \frac{1}{1-r^2} \end{pmatrix} .$$
The diagonal elements are the variances of the individual coefficient estimates, with variance inflation $1/(1-r^2)$. The covariance between $\hat\beta_1$ and $\hat\beta_2$ is opposite in sign to $r$. The Wald statistic for the hypothesis $\beta_1=0$ is then:
$$\frac{N \left( 1-r^2 \right) \hat\beta_1^2}{\hat\sigma^2},$$
with a corresponding result for $\beta_2$. Higher values of $r^2$ lower the values of test statistics on the individual coefficients, regardless of the sign of $r$. With more predictors, $1/(1-r^2)$ would be replaced by variance inflation factors in single-coefficient tests.
A Wald "chunk" test on the joint hypothesis that $\beta_1=\beta_2=0$ is based on the quadratic form between the vector $\begin{pmatrix}\hat\beta_1 & \hat\beta_2\end{pmatrix}$ and the inverse of the corresponding submatrix of the coefficient covariance matrix (last two columns and rows, in this example). The inverse of that submatrix is simply related to the corresponding submatrix of $X^TX$, giving:
$$\begin{pmatrix} \hat\beta_1 & \hat\beta_2\end{pmatrix} \frac{N}{\hat\sigma^2} \begin{pmatrix} 1 & r \\r&1 \end{pmatrix} \begin{pmatrix}\hat\beta_1 \\\hat\beta_2 \end{pmatrix}= \frac{N}{\hat\sigma^2} \left( \hat\beta_1^2 + 2 r \hat\beta_1 \hat\beta_2 + \hat\beta_2^2\right).$$
A positive correlation $r$ between $x_1$ and $x_2$, other things being equal, will increase the value of the "chunk" test statistic over the uncorrelated situation if $\hat\beta_1$ and $\hat\beta_2$ have the same sign, as will a negative correlation if the coefficient estimates have opposite signs.
*There's a generalized variance-inflation factor for other forms of regression; see this page.
