请看：
We have a standing invitation to visit them anytime.（正）
We have a standing invitation of visiting them anytime.（误）
他们邀请我们随时去他们家作客。
分析：invitation后接动词时只接to do不定式，表示“受邀做某事”；并不能接of doing形式。类似词语还有：decision， determination, willingness, resolution等，如：
I have an open invitation to visit my friend in Japan.
我在日本的朋友邀请我随时去看他。
I had an invitation to go and talk to the cadets.
我应邀去和军校学员谈话。
Philip refused his uncle’s invitation to go down to Blacktable.
飞利浦拒绝了他叔叔请他去布鲁克特布尔的邀请。
Elizabeth had a standing invitation to stay with her.
伊丽莎白受到邀请，可以随时去她家住。
The Syrians have not yet accepted an invitation to attend
叙利亚人还没有接到出席邀请。《柯林斯高阶英汉双解学习词典》
总感觉应该译为：
叙利亚人还没有明确接受出席邀请。（这里的接到邀请应该是get、receive）
We would appreciate confirmation of your refusal of our invitation to take part.
若您确认拒绝我们的邀请不能参加，我们将感激不尽。
当然invitation to+sth；也是地道用法：
It would be most ungracious and impolite to refuse a simple invitation to supper with him.
连与他共进晚餐这样的简单邀请都拒绝会显得非常失礼，没有教养。
I declined his invitation to dinner.
他要请我吃饭，我推却了。
John accepted the invitation to the party.
约翰接受了参加晚会的邀请。
To her surprise and joy, she received an invitation to that party.
她收到了赴宴的邀请，这让她又惊又喜。
They snubbed his invitation to a meeting of foreign ministers at the UN.
他们没有理睬他发出的出席联合国外交部长会议的邀请。
Eliza accepted Charles' invitation to a house party.
伊丽莎接受了查尔斯发出的参加家庭晚会的邀请。
He will have received an invitation to the wedding as well.
你肯定也会收到参加婚宴的邀请的。
She had wangled an invitation to the opening night.
她弄到了一张首映之夜的请柬。
顺便说一下：
Invitation of +人或组织机构团体；表示：某人或组织机构团体的邀请，如：
I visited Japan in 2023 at the invitation of the Foreign Minister.
应日本外相的邀请，我于2023年访问了日本。
He's understood to be there at the personal invitation of President Daniel Arap Moi.
据悉他是应总统丹尼尔·阿拉普·莫伊亲自邀请而到场的。
A concert was held at the invitation of the mayor.
在市长的邀请下举办了一场音乐会。