Vanguard ETF webpages have a "portfolio" section that lists their P/E, P/B and RoE.
Sadly, the P/E doesn't match P/B : RoE ratio.
eg for VOO (S&P 500 ETF)
PE is listed as 24.7, but PB 4.3 / RoE 24.61% comes to 15.
for VXF, PE is 18 but PB/RoE is 25.
Am I making some elementary calculation mistake here?
Even if taken on different dates in the quarter, that is a pretty significant discrepancy.

You maths looks good to me. RoE is exactly the Net Earnings / Book Value. The ratio can be calculated either as per share values or actual values.

So PB / RoE, as you write, or lets call it PB / (Return on Equity), can be expanded as:

(Price / Book) / (Net Earnings / Book)
Which is (Price / Book) * (Book / Net Earnings)
Which simplifies to (Price / Net Earnings) which is exactly the PE.

So PE = PB / (Return on Equity) as you observe.

As to why Vanguard report a discrepancy between the PE, PB and RoE, I do not know. However VOO tracks the S&P500 and from my sources, I have the following data:

S&P500
PE = 28 (price today, earnings trailing 12 months)
Price/Book = 4.9 (price today)

Both of these figures are a little different to what Vanguard is reporting for VOO which as I understand is exactly tracks the S&P500.

In any case, with my figures the RoE for the S&P500 can be calculated as Return on Equity = PB / PE = 4.9 / 28 = 0.175 = 17.5%.

It is not inuitive to think of Return on Equity as the PB / PE, however you reconcile it mentally with several sanity checks, such as:
1. If you double the earnings, the PE will become half.
2. If you double the earnings, Return on Equity (as it is PB / PE) will thus double. Checks out.
3. If you keep the earnings the same, but double the price, both PB and PE exactly double, and because Return on Equity = PB / PE, both numerator and denominator double and cancel out, so doubling the the price has no effect on Return on Equity. Checks out.

- Manlobbi