The fixed distance between the endpoints is essential for the question, because otherwise a simple rescaling of an arbitrary curve would work.

edit

the algorithm, that motivated this question is essentially based on realizing, that no point of the curve can lie outside the ellipse centered at $\left(\frac{\alpha}{2},0\right)$, foci $p$ at $\left(0,0\right)$ and $q$ at $\left(\alpha,0\right)$, for which the length of the semi-major axis equals $\frac{1}{2}$ and, $\sqrt{\left(\frac{1}{2}\right)^2-\left(\frac{\alpha}{2}\right)^2}$ for the semi-minor axis.
If the intermediate curve-point $r$ is chosen from the boundary of that ellipse, then the "length-stock" is used up and the algorithm terminates with a curve consisting of two line-segments and exact length $1$, joining $p$ and $q$ as demanded.

$\text{expand}$(Point $p$, Point $q$, Length $\ell$, Curve curve)

$\quad$Point $r\in \lbrace x\in\mathbb{R}^n\ |\ \| r-p \| + \|q-r\|\ \le \ell\rbrace$;
$\quad$Length $\ell_{pr}$ := $\|r-p\|$;
$\quad$Length $\ell_{rq}$ := $\|q-r\|$;
$\quad$Length $\Delta\ell$ := $\ell-\left(\ell_{pr}+\ell_{rq}\right)$;
$\quad$Scalar $a\in\left[0,1\right]$

$\quad$if (a < threshold)
$\quad\quad$curve.append($r-p$);
$\quad$else
$\quad\quad\text{expand}$($p$,$\ r$,$\ \ell_{pr}$+$a$ * $\Delta\ell$);

$\quad$if ($1-a$ < threshold)
$\quad\quad$curve.append($q-r$);
$\quad$else
$\quad\quad\text{expand}$($r$,$\ q$,$\ \ell_{rq}$+(1-$a$) * $\Delta\ell$);

<br/ > Some remarks:

The fixed distance between the endpoints is essential for the question, because otherwise a simple rescaling of an arbitrary curve would work.

Edit

Here are the promised details:

The algorithm, that motivated this question is essentially based on realizing, that no point of the curve can lie outside the ellipse centered at $\left(\frac{\alpha}{2},0\right)$, foci $p$ at $\left(0,0\right)$ and $q$ at $\left(\alpha,0\right)$, for which the length of the semi-major axis equals $\frac{1}{2}$ and, $\sqrt{\left(\frac{1}{2}\right)^2-\left(\frac{\alpha}{2}\right)^2}$ for the semi-minor axis.
If the intermediate curve-point $r$ is chosen from the boundary of that ellipse, then the "length-stock" is used up and the algorithm terminates with a curve consisting of two line-segments and exact length $1$, joining $p$ and $q$ as demanded.

$\text{expand}$(Point $p$, Point $q$, Length $\ell$, Curve curve)

$\quad$Point $r\in \lbrace x\in\mathbb{R}^n\ |\ \| r-p \| + \|q-r\|\ \le \ell\rbrace$;
$\quad$Length $\ell_{pr}$ := $\|r-p\|$;
$\quad$Length $\ell_{rq}$ := $\|q-r\|$;
$\quad$Length $\Delta\ell$ := $\ell-\left(\ell_{pr}+\ell_{rq}\right)$;
$\quad$Scalar $a\in\left[0,1\right]$

$\quad$if (a < threshold)
$\quad\quad$curve.append($r-p$);
$\quad$else
$\quad\quad\text{expand}$($p$,$\ r$,$\ \ell_{pr}$+$a$ * $\Delta\ell$);

$\quad$if ($1-a$ < threshold)
$\quad\quad$curve.append($q-r$);
$\quad$else
$\quad\quad\text{expand}$($r$,$\ q$,$\ \ell_{rq}$+(1-$a$) * $\Delta\ell$); 

Some remarks:

the algorithm, that motivated this question is essentially based on realizing, that no point of the curve can lie outside the ellipse centered at $\left(\frac{\alpha}{2},0\right)$, foci $p$ at $\left(0,0\right)$ and $q$ at $\left(\alpha,0\right)$, for which the length of the semi-major axis equals $\frac{1}{2}$ and, $\sqrt{\left(\frac{1}{2}\right)^2-\left(\frac{\alpha}{2}\right)^2}$ for the semi-minor axis.
If the intermediate curve-point $r$ is chosen from the boundary of that ellipse, then the "length-stock" is used up and the algorithm terminates with a curve consisting of two line-segments and exact length $1$, joining $p$ and $q$ as demanded.

Otherwise the length-stock is split up and assigned to two newly generated line-segments and the original problem of finding a curve of fixed length with endpoints at fixed has to be solved recursively for both segments separately.

Pseudo code:

$\text{expand}$(Point $p$, Point $q$, Length $\ell$, Curve curve)

$\quad$Point $r\in \lbrace x\in\mathbb{R}^n\ |\ \| r-p \| + \|q-r\|\ \le \ell\rbrace$;
$\quad$Length $\ell_{pr}$ := $\|r-p\|$;
$\quad$Length $\ell_{rq}$ := $\|q-r\|$;
$\quad$Length $\Delta\ell$ := $\ell-\left(\ell_{pr}+\ell_{rq}\right)$;
$\quad$Scalar $a\in\left[0,1\right]$

$\quad$if (a < threshold)
$\quad\quad$curve.append($r-p$);
$\quad$else
$\quad\quad\text{expand}$($p$,$\ r$,$\ \ell_{pr}$+$a$ * $\Delta\ell$);

$\quad$if ($1-a$ < threshold)
$\quad\quad$curve.append($q-r$);
$\quad$else
$\quad\quad\text{expand}$($r$,$\ q$,$\ \ell_{rq}$+(1-$a$) * $\Delta\ell$);

<br/ > Some remarks:

the pseudo code is aimed at full generality and also covers "degenerate" cases; those need to be ruled out by further checks. One such case is the collinearity of $p$, $\ q$ and $r$ with positive $\Delta\ell$.

Selecting $r$ from the mentioned elliptical regions with foci $p$ and $q$ can also be interpreted as chosing one of the intersection points of a circle around $p$ and $q$. That covers the algorithm of Matt F. as a special case.

The followup challenge is now to control further properties of the curve via taylored rules for selecting $r$ and distributing $\Delta\ell$ on each recursion level.
Or, play with the options to discover interesting curves and fractals.

Are algorithms already known, that generate (arbitrarily good approximations of) random curves, w.l.o.g. with unit length, and joining endpoints $(0,0)$ and $(\alpha,0)$ with $\alpha \lt1$ given?

The fixed distance between the endpoints is essential for the question, because otherwise a simple rescaling of an arbitrary curve would work.

Are algorithms already known, that generate (arbitrarily good approximations of) random curves, w.l.o.g. with unit length, and joining endpoints $(0,0)$ and $(\alpha,0)$ with $\alpha \lt1$ given?

The fixed distance between the endpoints is essential for the question, because otherwise a simple rescaling of an arbitrary curve would work.

edit

In view of the comments and the answer of Bjørn Kjos-Hanssen, I see the need for some clarification:

• By random curve of unit length connecting $(0,0)$ and $(\alpha,0)$, I mean a random sample from the space of all such curves; that means, that the algorithm should be capable to approximate every such curve to arbitrary precision with a finite number of steps.
  So "random" is not restricted to the appearance of the curve.

• Being able to generate Brownian Bridges is not sufficient, because I would like the algorithm to be able to generate curves (ideally in any $\mathbb{R}^n$) and not only functions.

So my apologies for not being precise enough.

I have used the formulation "are algorithms already known", because I have found one, that seems to be able to produce all those curves.

I will provide details in a later edit.