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it Xanathar's Guide to Everything, it states that when crafting an item:
"...a character needs to be proficient with the tools needed to craft an item and have access to the appropriate equipment." -XGtE, p. 128
If a character were to have proficiency with, for instance, smith's tools, would they be able to craft items regardless of whether or not they had access to a forge?
The answer to your question seems to be found in the quote you supplied:
"...a character needs to be proficient with the tools needed to craft an item and have access to the appropriate equipment." -XGTE, page 128
The use of the word "and" in this quote emphasises that a character needs both proficiency with tools and, also access to appropriate equipment.
While there to be some overlap between the two categories ('tools' are clearly at least part of what would be considered 'appropriate equipment'), if the two categories were supposed to be understood as synonymous, then that could have been expressed like this:
"...a character needs to be proficient with the tools needed to craft an item and have access to those tools."
As the text was not written this way, it seems logical to conclude that something extra, on top of simply 'tools', is intended to be covered by 'appropriate equipment'.
Therefore, a PC with proficiency in smith's tools, would still need a forge (and presumably an anvil) in order to craft items, not just their 'tools'. A forge, and an anvil, could sensibly be considered to be covered by the phrase appropriate equipment.
This makes sense, as a PC isn't expected to be able to smith with any less tools of the trade than a normal non-heroic NPC smith. They're not smithing magically.
Proficiency means that a PC is trained in smithing, and so any attempt they make to smith something is more likely to end positively (because they can add their proficiency bonus), than for a PC without equivalent training. Some DMs might not let a PC without proficiency attempt any smithing at all.
This doesn't mean that your PC has to maintain their own fully furnished smithy, and cover all of the upkeep costs incurred by owning a business while they're off adventuring (though they could). But, it does mean that during any downtime period they...
"possess the tools (and the skills) needed to pursue a trade"-PHB
...though they'd likely first need to make the acquaintance of the local smith and negotiate to use their forge, just like any other journeyman artisan.
