Does anything like <a href="fax:number">number</a> exist?

As we use

<a href="tel:number">number</a> or <a href="mailto:mailid">mailid</a>

for telephone number & email, does anything like <a href="fax:number">number</a> for fax exist?

Yes, the tel, fax and modem URL Schemes are discussed in https://www.ietf.org/rfc/rfc2806

2.3 "fax" URL scheme

> The URL syntax is formally described as follows (the definition reuses nonterminals from the above definition). For the basis of this syntax, see [RFC2303] and [RFC2304].

  fax-url          = fax-scheme ":" fax-subscriber
  fax-scheme       = "fax"
  fax-subscriber   = fax-global-phone / fax-local-phone
  fax-global-phone = "+" base-phone-number [isdn-subaddress]
                     [t33-subaddress] [post-dial]
                     *(area-specifier / service-provider /
                     future-extension)
  fax-local-phone  = 1*(phonedigit / dtmf-digit /
                     pause-character) [isdn-subaddress]
                     [t33-subaddress] [post-dial]
                     area-specifier
                     *(area-specifier / service-provider /
                     future-extension)
  t33-subaddress   = ";tsub=" 1*phonedigit

> The fax: URL is very similar to the tel: URL. The main difference is that in addition to ISDN subaddresses, telefaxes also have an another type of subaddress, see section 2.5.8.

Example:

<a href="fax:+358.555.1234567">+358.555.1234567</a>

However, as per the comments below, RFC 2806 was obsoleted by RFC 3966 and you should just use the tel URI scheme:

> The "tel" URI does not specify the call type, such as voice, fax, or data call, and does not provide the connection parameters for a data call. The type and parameters are assumed to be negotiated either in-band by the telephone device or through a signaling protocol such as SIP. This document obsoletes RFC 2806.

RFC 3966 was updated by RFC 5341 to formalize URI parameters and require them to be registered with IANA. I am including these here for the sake of completeness. They do not contain any additional information on fax devices.