PowerShell v2+, 60 bytes (but see the NB below)
A completely different approach, doing exactly what is requested by the challenge. ;-)
$a=2..$args[0]-join'*'|iex;"$a".length-"$a".Trim('0').length
Takes input $args and creates a dynamic range .., then -joins it together with * for multiplication, pipes that to iex (similar to eval) to compute the factorial. Store that in $a.
We then take the length of the string representation and subtract off the length of the string representation after we've passed it to .Trim('0') which will remove leading and trailing zeroes. Since there will never be leading zeroes, this gets us the result we want.
NB: The above will work for up to n=17 without a problem, as even though 17! is bigger than [Int]::MaxValue, since we didn't explicitly specify the casting for $a, PowerShell will dynamically convert it to [double] for us upon overflow. Yeah ... in PowerShell, [int] will overflow to [double] if not explicitly cast. No, I don't know why it goes to that rather than [int64].
However, at 18!, PowerShell will automatically convert the number to scientific notation when the implicit .ToString() is called (i.e., when encapsulating it within quotes), in order to preserve significant digits. We can get around that with explicit formatting instead -- $a="{0:0}"-f(2..$args[0]-join'*'|iex);$a.length-$a.Trim('0').length -- at 67 bytes. That will get us up to 20! before we start to run into loss of precision.
Thus, to handle an "arbitrary" input, we'll need to combine the above explicit formatting with the [bigint] datatype ...
PowerShell v2+, 80 bytes
$a="{0:0}"-f("[bigint]"+(2..$args[0]-join'*')|iex);$a.length-$a.Trim('0').length
This will prepend an explicit cast of [bigint] to the first number in the range 2, and in PowerShell the datatype on the left of the operator is used for output, so that [bigint] datatype will continue through the rest of the factorial calculation.
An few examples of the three methods at some key points, with some additional output text showing what's happening in the background -- the first output is the top code, the middle is with the explicit string formatting, and the bottom is with the [bigint] datatype.
PS C:\Tools\Scripts> .\golfing\zeroes-at-end-of-factorial.ps1 17
(Normal) 355687428096000 = 3
(Format) 355687428096000 = 3
(BigInt) 355687428096000 = 3
PS C:\Tools\Scripts> .\golfing\zeroes-at-end-of-factorial.ps1 18
(Normal) 6.402373705728E+15 = 0
(Format) 6402373705728000 = 3
(BigInt) 6402373705728000 = 3
PS C:\Tools\Scripts> .\golfing\zeroes-at-end-of-factorial.ps1 21
(Normal) 5.10909421717094E+19 = 0
(Format) 51090942171709400000 = 5
(BigInt) 51090942171709440000 = 4
PS C:\Tools\Scripts> .\golfing\zeroes-at-end-of-factorial.ps1 100
(Normal) 9.33262154439441E+157 = 0
(Format) 933262154439441000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000 = 143
(BigInt) 933262154439441526816992388562667004907159682643816214685929638952175999932299156089414639761565182862536979208272237582511852109168640000000000000000
00000000 = 24
PS C:\Tools\Scripts> .\golfing\zeroes-at-end-of-factorial.ps1 666
(Normal) Infinity = 0
(Format) Infinity = 0
(BigInt) 101063205684078149339082270812987645175758239832414541134042080735741380210369702298920280680149101204098980220355752703933970405713072930283454242384
016585642874066153029797241068282869939717688434251350949378748077490349338925526287834176188326189942648494465716169313138031111761957305152642332038964180541
081606760789306748325981681536460982866866274811038560365797328460484207809414155642770874534510059882948847250594907196772727091196506088520929434066550648022
642608335790150309778114083249701373807911277761571911620331754219999948922714475266708579675248268885046126373228453917614236582397369676453760327876932228670
885547506983568164371084614056976933006577541441308350104365957229945444651724282400214055514046429629100190143841467573055296491456926973403850076414055114364
283612861330473414734808609512385966092678846067118146921625221337465049955783174195059482714722569989641408869425126104519667256749553222882671938160611697400
311264211156133257350321296072971178199390387741639438171846476552757501425212904028323696392262434445697502405816736843180906854457725847298397943781807264821
360865009874936976105696120379126536366566469680224519996204004154443821032721047698220334845859609307929656956126740947391412413210205581149373619966878853487
232170536051130524871079644147921335454258357607659625021345466796883799602327316306909470042946710666392541958119313633986054565867362395523193239940480940410
876723200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000 = 165
n!would fit into your integer type! Well, maybe another time. \$\endgroup\$