It’s much easier to simultaneously construct $X_i$ and $Y_i$ having the desired properties,
by first letting $Y_i$ be i.i.d. Uniform$[0,1]$ and then taking $X_i = F^{-1}(Y_i)$. This is the basic method for generating random variables with arbitrary distributions.
The other direction, where you are first given $X_i$ and then asked to construct $Y_i$, is more difficult, but is still possible for all distributions. You just have to be careful with how you define $Y_i$.
Attempting to define $Y_i$ as $Y_i = F(X_i)$ fails to produce uniformly distributed $Y_i$ when $F$ has jump discontinuities. You have to spread the point masses in the distribution of $X_i$ across the the gaps created by the jumps.
Let $$D = \{x : F(x) \neq \lim_{z \to x^-} F(z)\}$$ denote the set of jump discontinuities of $F$. ($\lim_{z\to x^-}$ denotes the limit from the left. All distributions functions are right continuous, so the main issue is left discontinuities.)
Let $U_i$ be i.i.d. Uniform$[0,1]$ random variables, and define
$$Y_i =
\begin{cases}
F(X_i), & \text{if }X_i \notin D \\
U_i F(X_i) + (1-U_i) \lim_{z \to X_i^-} F(z), & \text{otherwise.}
\end{cases}
$$
The second part of the definition fills in the gaps uniformly.
The quantile function $F^{-1}$ is not a genuine inverse when $F$ is not 1-to-1. Note that if $X_i \in D$ then $F^{-1}(Y_i) = X_i$, because the pre-image of the gap is the corresponding point of discontinuity. For the continuous parts where $X_i \notin D$, the flat sections of $F$ correspond to intervals where $X_i$ has 0 probability so they don’t really matter when considering $F^{-1}(Y_i)$.
The second part of your question follows from similar reasoning after the first part which asserts that $X_i = F^{-1}(Y_i)$ with probability 1. The empirical CDFs are defined as
$$G_n(y) = \frac{1}{n} \sum_{i=1}^n 1_{\{Y_i \leq y\}}$$
$$F_n(x) = \frac{1}{n} \sum_{i=1}^n 1_{\{X_i \leq x\}}$$
so
$$
\begin{align}
G_n(F(x))
&= \frac{1}{n} \sum_{i=1}^n 1_{\{Y_i \leq F(x) \}}
= \frac{1}{n} \sum_{i=1}^n 1_{\{F^{-1}(Y_i) \leq x \}}
= \frac{1}{n} \sum_{i=1}^n 1_{\{X_i \leq x \}}
= F_n(x)
\end{align}
$$
with probability 1.
It should be easy to convince yourself that $Y_i$ has Uniform$[0,1]$ distribution by looking at pictures. Doing so rigorously is tedious, but can be done. We have to verify that $P(Y_i \leq u) = u$ for all $u \in (0,1)$. Fix such $u$ and let $x^* = \inf\{x : F(x) \geq u \}$ — this is just the value of quantile function at $u$. It’s defined this way to deal with flat sections. We’ll consider two separate cases.
First suppose that $F(x^*) = u$. Then
$$
Y_i \leq u
\iff Y_i \leq F(x^*)
\iff F(X_i) \leq F(x^*).
$$
Since $F$ is a non-decreasing function and $F(x^*) = u$,
$$
F(X_i) \leq F(x^*) \iff X_i \leq x^* .
$$
Thus,
$$
P[Y_i \leq u]
= P[X_i \leq x^*]
= F(x^*)
= u .
$$
Now suppose that $F(x^*) \neq u$. Then necessarily $F(x^*) > u$, and $u$ falls inside one of the gaps. Moreover, $x^* \in D$, because otherwise $F(x^*) = u$ and we have a contradiction.
Let $u^* = F(x^*)$ be the upper part of the gap. Then by the previous case,
$$
\begin{align}
P[Y_i \leq u]
&= P[Y_i \leq u^*] - P[u < Y_i \leq u^*]\\
&= u^* - P[u < Y_i \leq u^*].
\end{align}
$$
By the way $Y_i$ is defined, $P(Y_i = u^*) = 0$ and
$$
\begin{align}
P[u < Y_i \leq u^*]
&= P[u < Y_i < u^*] \\
&= P[u < Y_i < u^* , X_i = x^*] \\
&= u^* - u .
\end{align}
$$
Thus, $P[Y_i \leq u] = u$.
If the counts are all likely to be large, the main potential issue I see here is the variance function, since you don't have anything that scales the biomass to an actual count. It's like having a noisy scaled count without knowing the scaling factor. That may not be such an issue with the negative binomial as it is with the Poisson, though.
If you have some atoms of probability but the data are otherwise continuous you have a mixed distribution (a mixture of continuous and discrete); when the only atom is at zero, it's sometimes called a zero-inflated continuous distribution.
Zero-inflated gamma and Zero-inflated lognormal distributions are commonly used; either might suit your case. Typical models include zero-inflated and hurdle models (yes, the term zero-inflated is overloaded). These are often applied to discrete data (e.g. for otherwise Poisson data you have Poisson hurdle and zero-inflated Poisson, or ZIP models), where the models are different in how they treat zeros, but the distinction is less clearly drawn for continuous models; but if I used different variables to model the zeros from the model for the continuous part I'd tend to call it a hurdle model rather than zero-inflated. If I used the same form of linear predictor (but with different betas), or if I had a constant probability of zero, I'd probably call it a zero-inflated model -- however, I'm not an expert on such models, so you may be better off following other people's way of dividing up models for continuous zero-inflated data.
There are some posts on our site relating to zero-inflated gamma models and other zero-inflated distributions, and on continuous zero-inflated and/or hurdle models.
On this page, Sean Anderson talks about gamma hurdle models and specifically mentions its use for modelling biomass.
Portion of older answer given under the original post (which stated the distribution was continuous):
I'd be inclined to model it as a gamma; it's continuous, and it arguably has roughly similar properties to the negative binomial.
Is there a particular reason you need the negative binomial?
Best Answer
Wikipedia has a list of distributions supported on an interval
Leaving aside mixtures and 0-inflated and 0-1 inflated cases (though you should definitely be aware of all of those if you model data on the unit interval), which ones are common would be hard to establish (it will vary across application areas for example), but the beta family, and the triangular, and the truncated normal would probably be the main candidates as they seem to be used in a variety of situations.
Each of them can be defined on (0,1) and can be skewed either direction.
One example of each is shown here:
That they're often used doesn't imply they'll be suitable for whatever situation you're in, though. Model choice should be based on a number of considerations, but where possible, theoretical understanding and practical subject area knowledge are both important.
You should get away from worrying about "best", and focus on "sufficient/adequate for the present purpose". No simple distribution such as the ones I mentioned will really be a perfect description of real data ("all models are wrong..."), and what might be fine for one purpose ("... some are useful") may be inadequate for some other purpose.
Edit to address information in comments:
If you have exact zeros (or exact ones, or both), then you will need to model the probability of those 0's and use a mixture distribution (a 0-inflated distribution if you can have exact 0's) -- shouldn't use a continuous distribution.
It's not really all that hard to deal with simple mixtures. You'll no longer have a density but the cdf is not much more effort to write down or evaluate than it would be in the continuous case; similarly quantiles are not much more effort either; means and variances are almost as readily calculated as before; and they're easy to simulate from.
Taking an existing continuous distribution on the unit interval and adding a proportion of zeros (and/or ones) is on the whole a pretty convenient way to model proportions that are mostly continuous but can be 0 or 1.