Is there a statistical distribution whose values are bounded $[-1,1]$ and sum to 1? The Dirichlet distribution contains values that are bounded $[0,1]\in \mathbb{R}$ and sum to $1$. Is there a parametric distribution or similar method whose values do the same but reach as low as $-1$?
Parallel discussion of the code
 A: Scaling a Dirichlet distribution
If you want a variable that is distributed like a Dirichlet distributed variable but with a different range then you can scale and shift (transform the variable). This is effectively rescaling the axes.
To get from $[0,1]$ to $[-1,1]$ you can multiply by 2 and subtract 1. That is, your new variable $Y$ can be based on a regular Dirichlet distributed variable $X$ by the transformation
$$Y = 2X -1$$
(Where the transformation is done for each of the components, that is for every $y_i$ you compute $y_i = 2x_i-1$)

The probability density function will scale similarly but with an additional scaling factor (the density is less when you spread out over a larger range).
So the regular Dirichlet distributed variable $X$ has the density distribution $f_X$:
$$f_X(\mathbf{x}) = \frac{1}{B(\boldsymbol{\alpha})} \prod_{i=1}^K x_i^{\alpha_i-1}$$
and the variable $Y = 2X-1$ has this density distribution $f_Y$:
$$f_Y(\mathbf{y}) = \frac{1}{2^K} f_X \left(\frac{\mathbf{y}+1}{2}\right) = \frac{1}{B(\boldsymbol{\alpha})2^K} \prod_{i=1}^K \left(\frac{y_i+1}{2}\right)^{\alpha_i-1}$$
where $B(\mathbf{\boldsymbol{\alpha}}) =\prod_{i=1}^K \frac{\Gamma(\alpha_i)}{\Gamma(\sum_{i=1}^K \alpha_i)}$

So you do not need to change anything to $\alpha$. The transformation only requires scaling and shifting the axes (which also includes a scaling of the density by a factor $1/2^K$).
Whatever $\alpha$ needs to be will depend on your application.

When there is a constraint

Is there a statistical distribution whose values are bounded [−1,1]
and sum to 1?

Note: This transformation by scaling the axis is not always generally possible in case of your additional constraint.
Your additional condition requires $$\sum_{i=1}^n y_i = \sum_{i=1}^n (a + b x_i) = an + b \sum_{i=1}^n x_i = 1$$ and this only holds when $n = \frac{1-b}{a}$. With our straightforward transformation $a=-1$ and $b=2$ it does not hold. We need to use instead $a=1$ and $b=-2$, and then it will only work for a Dirichlet distribution with $n=3$.
The figure below shows this

The red plane is the domain of the 'regular' Dirichlet distribution.
The green plane is when you apply the transformation $y_i = 2x_i -1$, but then you do not get anymore that the variables sum up to 1. Instead the variables will sum up to -1.
The blue plane $y_i = 1 - 2 x_i$ will give you a transformation such that the sum is still 1.
A homogeneous distribution
Based on your stackoverflow question it seems that you are not looking for a distribution like the Dirichlet distribution, but you are looking for a homogeneous distribution (a special case of the Dirichlet distribution when all $\alpha_i =1$), where the pdf equals some constant $f(\mathbf{x}) = c$.
You can do this by rejection sampling or by an iterative computation of the coordinates $x_i$ where conditional/marginal distributions $f(x_i|x_1,x_2,\dots,x_{i-1})$ can be derived from rescaled and truncated versions of the Irwin Hall distribution. It is explained in the answer to your stackoverflow question.
A: If you really need the variables to sum to one, you could "force it" by dividing by the sum. That is, if $X_1, X_2, \cdots X_n$ are random variables, then the RVs
$$Z_i = \frac{X_i}{\sum_{i=1}^n X_i}$$
have the property that $\sum_{i=1}^nZ_i = 1$ (so long as $\sum X_i \neq 0$). This is easy to show.
$$\sum_{j=1}^n Z_j = \sum_{j=1}^n \frac{X_j}{\sum_{i=1}^n X_i} = \frac{1}{\sum_{i=1}^n X_i}\sum_{j=1}^n X_j = 1$$

N <- 10000
x <- 1 - 2*rbeta(N, 3, 3)
z <- x/sum(x)
w <- -1 + 2*(z-min(z))/(max(z) - min(z))
par(mfrow=c(1,2))
hist(x)
hist(z)


