If audit was conducted 365 times, what is the probability that it would pass 365*0.9 times? The probability can be calculated from Binomial distribution $B(k|365,p)$. The problem is that $p$ is unknown. So, we need a trick.
First, we estimate it by conducting $n$ trials and get $\hat p=\frac{m+1}{n+2}$, where $m$ was the number of times audit passed in $n$ trials. Here, I'm using the Laplace's rule of succession. The reason is that if you have $n$ successes then usual MLE estimate gives $\hat p=1$ and it messes up the bootstrapping step.
Second, we need a sampling distribution of $\hat p$. We can either an approximation or an "exact" one. There are many approximations, e.g.
using Wald method we get the sampling distribution:
$$f(\tilde p)=\hat p+ z\sqrt{\hat p(1-\hat p)/n},$$
where z is the standard normal.
The "exact" method isn't really exact but it's a bootstrapping from $\hat p$: we plug it into the binomial distribution, generate $n$ trials and get $\tilde p=\tilde m/n$, where $\tilde m$ is number of passes. This is what "exact" method is based on.
Third, we convolve $f(\hat p)$, either approximation or "exact", with the binomial distribution:
$$k\sim \int_{0}^1 B(k|365,\hat p)f(\hat p)d \hat p$$
Where $k$ is the number of times passed in 365 audits.
Once you got the distribution, perhaps numerically, you can estimate the probability $Pr[k>365*0.9]$ and get the probability that you were looking for.
Now, you should see that your question doesn't have a unique answer. In other words, you pick any $n$ and produce a probability that auditee would have passed 90% of time in daily audits during the year. The only things that will change is the power of the test, it will increase with $n$, but you didn't specify the required power.
Example: suppose you conducted n=46 audits and 42 of them passed. This means $\hat p=0.896$, and the sampling distribution is $f(\hat p)=\mathcal N(0.896,0.045)$ in Wald method or in "exact" method you get the corresponding Binomial distribution. Monte Carlo simulation with 1000 trials produces the probability of passing 329 or more time in 365 trials is $\pi\approx 47\pm 1.6$% in Wald and $\pi\approx 35\pm 1.5$% in "exact" method. The Monte Carlo uncertainty is $\sqrt{\pi*(1-\pi)/1000}\approx 1.5$%. Here's a cumulative histogram for Wald and "exact" methods:
VERY IMPORTANT CASE: ongoing audit
The previous discussion was based on the notion that you did $n$ trials this year, and planning for the next year audit. What if we're planning the remaining audit? In other words $n$ is a part of 365 days of audit. So we cunducted $n$ audits and now are thinking how many more we need before we're confident that auditee would pass 90% of total 365 audits?
In this case you have to make an important adjustment to the step 3 in the previous discussion. You only generate 365-$n$ trials in Monte Carlo if you're using this method. Here's what I got with "exact" method for this case and the same n=46 and m=43. The estimate of probability is the same $\hat p=0.896$. So we run Monte Carlo with this setting but generate only passes for 365-46 trails and add the passed to earlier 42 passed. Then we build the cumulative distribution of total passes. In this case the probability of 90% passes in total 365 trials changes a bit: $\pi\approx 38\pm1.5$%. Here's a histogram:
This modification doesn't make a lot of difference for low initial trial numbers $n$. However, as $n$ grows it becomes very important. Consider the case where you made already $n=364$ trials with 328 successes. In this case the probability of getting 90% successes is equal $\hat p$, because we have essentially a Bernoulli experiment at hand at this point. Unlike the case where you're planning for the next year, where you didn't run any audit at all yet.
This is how you do it in Excel.
First you calculate the estimated $\hat p$ in cell F6 using the Laplace formula. Then you bootstrap the sampling distribution of $\tilde p$ in column B:
Then for each realization of $\tilde p$ in column B you simulate remianing trials and show obtained passes k in column C:
Finally, you count in how many simulations you passed the required mark of 90% in 365 audits:
